Term functions and polynomial functions in hyperstructure...
Transcript of Term functions and polynomial functions in hyperstructure...
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Term functions and polynomial functions inhyperstructure theory
Ioan Purdea and Cosmin Pelea
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 2: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/2.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Multialgebra
Let τ = (nγ)γ<o(τ) be a sequence of nonnegative integers (o(τ) isan ordinal) and for any γ < o(τ), let fγ be a symbol of an nγ-ary(multi)operation.
A multialgebra A of type τ consists in a set A and a family ofmultioperations (fγ)γ<o(τ), where
fγ : Anγ → P∗(A)
is the nγ-ary multioperation which corresponds to the symbol fγ .
Denote byP(n)(τ) = (P(n)(τ), (fγ)γ<o(τ))
the algebra of the n-ary terms (of type τ).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I A multialgebra A = (fγ)γ<o(τ) can be seen as a relational system(A, (rγ)γ<o(τ)), where rγ is the nγ + 1-ary relation defined by
(a0, . . . , anγ−1, anγ ) ∈ rγ ⇔ anγ ∈ fγ(a0, . . . , anγ−1).
I The algebra of the nonempty subsets of a multialgebra (H. E.Pickett):
A multialgebra A determines a universal algebra P∗(A) on P∗(A)defining for any A0, . . . ,Anγ−1 ∈ P∗(A),
fγ(A0, ...,Anγ−1) =⋃{fγ(a0, ..., anγ−1) | ai ∈ Ai , i = 0, nγ − 1}.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 4: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/4.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I A multialgebra A = (fγ)γ<o(τ) can be seen as a relational system(A, (rγ)γ<o(τ)), where rγ is the nγ + 1-ary relation defined by
(a0, . . . , anγ−1, anγ ) ∈ rγ ⇔ anγ ∈ fγ(a0, . . . , anγ−1).
I The algebra of the nonempty subsets of a multialgebra (H. E.Pickett):
A multialgebra A determines a universal algebra P∗(A) on P∗(A)defining for any A0, . . . ,Anγ−1 ∈ P∗(A),
fγ(A0, ...,Anγ−1) =⋃{fγ(a0, ..., anγ−1) | ai ∈ Ai , i = 0, nγ − 1}.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I A multialgebra A = (fγ)γ<o(τ) can be seen as a relational system(A, (rγ)γ<o(τ)), where rγ is the nγ + 1-ary relation defined by
(a0, . . . , anγ−1, anγ ) ∈ rγ ⇔ anγ ∈ fγ(a0, . . . , anγ−1).
I The algebra of the nonempty subsets of a multialgebra (H. E.Pickett):
A multialgebra A determines a universal algebra P∗(A) on P∗(A)defining for any A0, . . . ,Anγ−1 ∈ P∗(A),
fγ(A0, ...,Anγ−1) =⋃{fγ(a0, ..., anγ−1) | ai ∈ Ai , i = 0, nγ − 1}.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 6: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/6.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Identities
Let q, r ∈ P(n)(τ) and let A be a multialgebra of type τ .
I (Strong) identity : The n-ary (strong) identity
q = r
is satisfied on A if
q(a0, . . . , an−1) = r(a0, . . . , an−1), ∀a0, . . . , an−1 ∈ A.
I Weak identity (T. Vougiouklis): The weak identity
q ∩ r 6= ∅
is satisfied on A if
q(a0, . . . , an−1) ∩ r(a0, . . . , an−1) 6= ∅, ∀a0, . . . , an−1 ∈ A.
(q and r denotes the term functions induced by q and r on P∗(A).)
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Identities
Let q, r ∈ P(n)(τ) and let A be a multialgebra of type τ .
I (Strong) identity : The n-ary (strong) identity
q = r
is satisfied on A if
q(a0, . . . , an−1) = r(a0, . . . , an−1), ∀a0, . . . , an−1 ∈ A.
I Weak identity (T. Vougiouklis): The weak identity
q ∩ r 6= ∅
is satisfied on A if
q(a0, . . . , an−1) ∩ r(a0, . . . , an−1) 6= ∅, ∀a0, . . . , an−1 ∈ A.
(q and r denotes the term functions induced by q and r on P∗(A).)
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 8: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/8.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Identities
Let q, r ∈ P(n)(τ) and let A be a multialgebra of type τ .
I (Strong) identity : The n-ary (strong) identity
q = r
is satisfied on A if
q(a0, . . . , an−1) = r(a0, . . . , an−1), ∀a0, . . . , an−1 ∈ A.
I Weak identity (T. Vougiouklis): The weak identity
q ∩ r 6= ∅
is satisfied on A if
q(a0, . . . , an−1) ∩ r(a0, . . . , an−1) 6= ∅, ∀a0, . . . , an−1 ∈ A.
(q and r denotes the term functions induced by q and r on P∗(A).)
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 9: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/9.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Identities
Let q, r ∈ P(n)(τ) and let A be a multialgebra of type τ .
I (Strong) identity : The n-ary (strong) identity
q = r
is satisfied on A if
q(a0, . . . , an−1) = r(a0, . . . , an−1), ∀a0, . . . , an−1 ∈ A.
I Weak identity (T. Vougiouklis): The weak identity
q ∩ r 6= ∅
is satisfied on A if
q(a0, . . . , an−1) ∩ r(a0, . . . , an−1) 6= ∅, ∀a0, . . . , an−1 ∈ A.
(q and r denotes the term functions induced by q and r on P∗(A).)
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 10: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/10.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
The tools
I The algebra P(n)P∗(A)(P∗(A)) of the n-ary polynomial functions
of the universal algebra P∗(A).
I The algebra P(n)A (P∗(A)) which is the subalgebra of
P(n)P∗(A)(P∗(A)) generated by
{cna | a ∈ A} ∪ {eni | i ∈ {0, . . . , n − 1}},
where cna , eni : P∗(A)n → P∗(A) are defined by
cna (A0, . . . ,An−1) = {a} and eni (A0, . . . ,An−1) = Ai .
I The algebra P(n)(P∗(A)) of the n-ary term functions on
P∗(A), which is the subalgebra of P(n)A (P∗(A)) generated by
{eni | i ∈ {0, . . . , n − 1}}.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 11: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/11.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
The tools
I The algebra P(n)P∗(A)(P∗(A)) of the n-ary polynomial functions
of the universal algebra P∗(A).
I The algebra P(n)A (P∗(A)) which is the subalgebra of
P(n)P∗(A)(P∗(A)) generated by
{cna | a ∈ A} ∪ {eni | i ∈ {0, . . . , n − 1}},
where cna , eni : P∗(A)n → P∗(A) are defined by
cna (A0, . . . ,An−1) = {a} and eni (A0, . . . ,An−1) = Ai .
I The algebra P(n)(P∗(A)) of the n-ary term functions on
P∗(A), which is the subalgebra of P(n)A (P∗(A)) generated by
{eni | i ∈ {0, . . . , n − 1}}.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 12: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/12.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
The tools
I The algebra P(n)P∗(A)(P∗(A)) of the n-ary polynomial functions
of the universal algebra P∗(A).
I The algebra P(n)A (P∗(A)) which is the subalgebra of
P(n)P∗(A)(P∗(A)) generated by
{cna | a ∈ A} ∪ {eni | i ∈ {0, . . . , n − 1}},
where cna , eni : P∗(A)n → P∗(A) are defined by
cna (A0, . . . ,An−1) = {a} and eni (A0, . . . ,An−1) = Ai .
I The algebra P(n)(P∗(A)) of the n-ary term functions on
P∗(A), which is the subalgebra of P(n)A (P∗(A)) generated by
{eni | i ∈ {0, . . . , n − 1}}.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 13: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/13.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Our purpose ...
... is to provide various samples to show how we can use thesetools when we are dealing with:
I the submultialgebras of A;
I multialgebra homomorphisms;
I factor multialgebras;
I the ideal equivalences of A;
• those equivalences for which the factor multialgebra of A isa universal algebra;
• the fundamental relation of A;
I direct products of multialgebras;
I direct limits of direct systems of multialgebras.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 14: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/14.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Our purpose ...
... is to provide various samples to show how we can use thesetools when we are dealing with:
I the submultialgebras of A;
I multialgebra homomorphisms;
I factor multialgebras;
I the ideal equivalences of A;
• those equivalences for which the factor multialgebra of A isa universal algebra;
• the fundamental relation of A;
I direct products of multialgebras;
I direct limits of direct systems of multialgebras.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 15: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/15.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Our purpose ...
... is to provide various samples to show how we can use thesetools when we are dealing with:
I the submultialgebras of A;
I multialgebra homomorphisms;
I factor multialgebras;
I the ideal equivalences of A;
• those equivalences for which the factor multialgebra of A isa universal algebra;
• the fundamental relation of A;
I direct products of multialgebras;
I direct limits of direct systems of multialgebras.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 16: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/16.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Our purpose ...
... is to provide various samples to show how we can use thesetools when we are dealing with:
I the submultialgebras of A;
I multialgebra homomorphisms;
I factor multialgebras;
I the ideal equivalences of A;
• those equivalences for which the factor multialgebra of A isa universal algebra;
• the fundamental relation of A;
I direct products of multialgebras;
I direct limits of direct systems of multialgebras.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 17: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/17.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Our purpose ...
... is to provide various samples to show how we can use thesetools when we are dealing with:
I the submultialgebras of A;
I multialgebra homomorphisms;
I factor multialgebras;
I the ideal equivalences of A;
• those equivalences for which the factor multialgebra of A isa universal algebra;
• the fundamental relation of A;
I direct products of multialgebras;
I direct limits of direct systems of multialgebras.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 18: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/18.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Our purpose ...
... is to provide various samples to show how we can use thesetools when we are dealing with:
I the submultialgebras of A;
I multialgebra homomorphisms;
I factor multialgebras;
I the ideal equivalences of A;
• those equivalences for which the factor multialgebra of A isa universal algebra;
• the fundamental relation of A;
I direct products of multialgebras;
I direct limits of direct systems of multialgebras.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 19: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/19.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Our purpose ...
... is to provide various samples to show how we can use thesetools when we are dealing with:
I the submultialgebras of A;
I multialgebra homomorphisms;
I factor multialgebras;
I the ideal equivalences of A;
• those equivalences for which the factor multialgebra of A isa universal algebra;
• the fundamental relation of A;
I direct products of multialgebras;
I direct limits of direct systems of multialgebras.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 20: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/20.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Our purpose ...
... is to provide various samples to show how we can use thesetools when we are dealing with:
I the submultialgebras of A;
I multialgebra homomorphisms;
I factor multialgebras;
I the ideal equivalences of A;
• those equivalences for which the factor multialgebra of A isa universal algebra;
• the fundamental relation of A;
I direct products of multialgebras;
I direct limits of direct systems of multialgebras.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 21: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/21.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
In an overwhelming number of situations we apply term functions
or polynomial functions from P(n)A (P∗(A)) to one element sets.
⇒ Question: Is it necessary to redefine term functions andpolynomial functions as some sort of special multioperationsfor these special situations?
Our opinion: No, since from a certain step of theirconstruction we operate sets, not elements. (Of course, theyare multioperations, but ignoring the fact that we are workingin P∗(A) can be dangerous.)
Example
For a hypergroupoid (H, ◦) and a, b, c ∈ H, the nonempty subsets
a ◦ (b ◦ c), (a ◦ b) ◦ (a ◦ c)
are, actually, compositions of sets.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 22: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/22.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
In an overwhelming number of situations we apply term functions
or polynomial functions from P(n)A (P∗(A)) to one element sets.
⇒ Question: Is it necessary to redefine term functions andpolynomial functions as some sort of special multioperationsfor these special situations?
Our opinion: No, since from a certain step of theirconstruction we operate sets, not elements. (Of course, theyare multioperations, but ignoring the fact that we are workingin P∗(A) can be dangerous.)
Example
For a hypergroupoid (H, ◦) and a, b, c ∈ H, the nonempty subsets
a ◦ (b ◦ c), (a ◦ b) ◦ (a ◦ c)
are, actually, compositions of sets.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 23: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/23.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
SubmultialgebrasLet A = (A, (fγ)γ<o(τ)) be a multialgebra.
A subset B ⊆ A is a submultialgebra of A if
fγ(b0, . . . , bnγ−1) ⊆ B, ∀γ < o(τ), ∀b0, . . . , bnγ−1 ∈ B.
Theorem (Pickett)
A subset B ⊆ A is a submultialgebra of A if and only if P∗(B) is asubalgebra of P∗(A).
⇒ If B is a submultialgebra of A, n ∈ N, p ∈ P(n)(P∗(A)) andb0, . . . , bn−1 ∈ B then
p(b0, . . . , bn−1) ⊆ B.
⇒ (Breaz, Pelea) If X ⊆ A then a ∈ 〈X 〉 if and only if there existn ∈ N, p ∈ P(n)(P∗(A)) and x0, . . . , xn−1 ∈ X such that
a ∈ p(x0, . . . , xn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 24: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/24.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
SubmultialgebrasLet A = (A, (fγ)γ<o(τ)) be a multialgebra.
A subset B ⊆ A is a submultialgebra of A if
fγ(b0, . . . , bnγ−1) ⊆ B, ∀γ < o(τ), ∀b0, . . . , bnγ−1 ∈ B.
Theorem (Pickett)
A subset B ⊆ A is a submultialgebra of A if and only if P∗(B) is asubalgebra of P∗(A).
⇒ If B is a submultialgebra of A, n ∈ N, p ∈ P(n)(P∗(A)) andb0, . . . , bn−1 ∈ B then
p(b0, . . . , bn−1) ⊆ B.
⇒ (Breaz, Pelea) If X ⊆ A then a ∈ 〈X 〉 if and only if there existn ∈ N, p ∈ P(n)(P∗(A)) and x0, . . . , xn−1 ∈ X such that
a ∈ p(x0, . . . , xn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 25: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/25.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
SubmultialgebrasLet A = (A, (fγ)γ<o(τ)) be a multialgebra.
A subset B ⊆ A is a submultialgebra of A if
fγ(b0, . . . , bnγ−1) ⊆ B, ∀γ < o(τ), ∀b0, . . . , bnγ−1 ∈ B.
Theorem (Pickett)
A subset B ⊆ A is a submultialgebra of A if and only if P∗(B) is asubalgebra of P∗(A).
⇒ If B is a submultialgebra of A, n ∈ N, p ∈ P(n)(P∗(A)) andb0, . . . , bn−1 ∈ B then
p(b0, . . . , bn−1) ⊆ B.
⇒ (Breaz, Pelea) If X ⊆ A then a ∈ 〈X 〉 if and only if there existn ∈ N, p ∈ P(n)(P∗(A)) and x0, . . . , xn−1 ∈ X such that
a ∈ p(x0, . . . , xn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Homomorphisms
Let A = (A, (fγ)γ<o(τ)) and B = (B, (fγ)γ<o(τ)) be multialgebrasof type τ and h : A→ B.
I h is a multialgebra homomorphism from A into B if for anyγ < o(τ) and a0, . . . , anγ−1 ∈ A,
h(fγ(a0, . . . , anγ−1)) ⊆ fγ(h(a0), . . . , h(anγ−1)).
I h is an ideal homomorphism if for any γ < o(τ) anda0, . . . , anγ−1 ∈ A,
h(fγ(a0, . . . , anγ−1)) = fγ(h(a0), . . . , h(anγ−1)).
I the isomorphisms are the bijective ideal homomorphisms.
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Homomorphisms
Let A = (A, (fγ)γ<o(τ)) and B = (B, (fγ)γ<o(τ)) be multialgebrasof type τ and h : A→ B.
I h is a multialgebra homomorphism from A into B if for anyγ < o(τ) and a0, . . . , anγ−1 ∈ A,
h(fγ(a0, . . . , anγ−1)) ⊆ fγ(h(a0), . . . , h(anγ−1)).
I h is an ideal homomorphism if for any γ < o(τ) anda0, . . . , anγ−1 ∈ A,
h(fγ(a0, . . . , anγ−1)) = fγ(h(a0), . . . , h(anγ−1)).
I the isomorphisms are the bijective ideal homomorphisms.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Homomorphisms
Let A = (A, (fγ)γ<o(τ)) and B = (B, (fγ)γ<o(τ)) be multialgebrasof type τ and h : A→ B.
I h is a multialgebra homomorphism from A into B if for anyγ < o(τ) and a0, . . . , anγ−1 ∈ A,
h(fγ(a0, . . . , anγ−1)) ⊆ fγ(h(a0), . . . , h(anγ−1)).
I h is an ideal homomorphism if for any γ < o(τ) anda0, . . . , anγ−1 ∈ A,
h(fγ(a0, . . . , anγ−1)) = fγ(h(a0), . . . , h(anγ−1)).
I the isomorphisms are the bijective ideal homomorphisms.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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The mapping h induces the mapping
h∗ : P∗(A)→ P∗(B), h∗(X ) = h(X ) = {h(x) | x ∈ X}.
Theorem (Pickett)
h is a multialgebra ideal homomorphism if and only if h∗ is auniversal algebra homomorphism.
⇒ If h : A→ B is an ideal homomorphism, n ∈ N, p ∈ P(n)(τ),and a0, . . . , an−1 ∈ A then
h(p(a0, . . . , an−1)) = p(h(a0), . . . , h(an−1)).
I If h : A→ B is a multialgebra homomorphism, n ∈ N,p ∈ P(n)(τ), and a0, . . . , an−1 ∈ A then
h(p(a0, . . . , an−1)) ⊆ p(h(a0), . . . , h(an−1)).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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The mapping h induces the mapping
h∗ : P∗(A)→ P∗(B), h∗(X ) = h(X ) = {h(x) | x ∈ X}.
Theorem (Pickett)
h is a multialgebra ideal homomorphism if and only if h∗ is auniversal algebra homomorphism.
⇒ If h : A→ B is an ideal homomorphism, n ∈ N, p ∈ P(n)(τ),and a0, . . . , an−1 ∈ A then
h(p(a0, . . . , an−1)) = p(h(a0), . . . , h(an−1)).
I If h : A→ B is a multialgebra homomorphism, n ∈ N,p ∈ P(n)(τ), and a0, . . . , an−1 ∈ A then
h(p(a0, . . . , an−1)) ⊆ p(h(a0), . . . , h(an−1)).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I The mappingsA 7→ P∗(A), h 7→ h∗
define a covariant functor from the category of themultialgebras of type τ having the ideal homomorphisms asmorphisms into the category of the universal algebras of typeτ .
I QUESTION 1: Can we use these mappings or this functor toestablish other connections between universal algebras andhyperstructure theory which can provide interesting and usefulresults for both theories, especially for hyperstructure theory?
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I The mappingsA 7→ P∗(A), h 7→ h∗
define a covariant functor from the category of themultialgebras of type τ having the ideal homomorphisms asmorphisms into the category of the universal algebras of typeτ .
I QUESTION 1: Can we use these mappings or this functor toestablish other connections between universal algebras andhyperstructure theory which can provide interesting and usefulresults for both theories, especially for hyperstructure theory?
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Factor multialgebras
Let A = (A, (fγ)γ<o(τ)) be a multialgebra of type τ and let ρ be anequivalence relation on A. Let ρ〈x〉 be the class of x modulo ρ, and
A/ρ = {ρ〈x〉 | x ∈ A}.
Defining for each γ < o(τ),
fγ(ρ〈a0〉, ..., ρ〈anγ−1〉) = {ρ〈b〉|b ∈ fγ(b0, ..., bnγ−1), aiρbi , i = 0, nγ − 1},
one obtains a multialgebra A/ρ on A/ρ called the factormultialgebra of A modulo ρ.
I G. Gratzer proved that any multialgebra is a factor of auniversal algebra modulo an equivalence relation.
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Factor multialgebras
Let A = (A, (fγ)γ<o(τ)) be a multialgebra of type τ and let ρ be anequivalence relation on A. Let ρ〈x〉 be the class of x modulo ρ, and
A/ρ = {ρ〈x〉 | x ∈ A}.
Defining for each γ < o(τ),
fγ(ρ〈a0〉, ..., ρ〈anγ−1〉) = {ρ〈b〉|b ∈ fγ(b0, ..., bnγ−1), aiρbi , i = 0, nγ − 1},
one obtains a multialgebra A/ρ on A/ρ called the factormultialgebra of A modulo ρ.
I G. Gratzer proved that any multialgebra is a factor of auniversal algebra modulo an equivalence relation.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Since the canonical map A→ A/ρ is a multialgebrahomomorphism, for any n ∈ N, p ∈ P(n)(τ) and a0, . . . , an−1 ∈ A,
{ρ〈a〉 | a ∈ (p)P∗(A)(a0, . . . , an−1)} ⊆ (p)P∗(A/ρ)(ρ〈a0〉, . . . , ρ〈an−1〉)
This inclusion holds if we replace (p)P∗(A) and (p)P∗(A/ρ) by
p ∈ P(n)A (P∗(A)) and p′ ∈ P
(n)A/ρ(P∗(A/ρ)) respectively, where the
polynomial function p′ corresponding to p is given as follows:
(i) if p = cna , then p′ = cnρ〈a〉;
(ii) if p = eni = (xi )P∗(A), then p′ = eni = (xi )P∗(A/ρ);
(iii) if p = fγ(p0, . . . , pnγ−1) and p′0, . . . , p′nγ−1 ∈ P(n)A/ρ(P∗(A/ρ))
are the polynomial functions which correspond to p0, . . . ,
pnγ−1 ∈ P(n)A (P∗(A)) respectively, then
p′ = fγ(p′0, . . . , p′nγ−1).
So, we can write
{ρ〈a〉 | a ∈ p(a0, . . . , an−1)} ⊆ p′(ρ〈a0〉, . . . , ρ〈an−1〉).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Since the canonical map A→ A/ρ is a multialgebrahomomorphism, for any n ∈ N, p ∈ P(n)(τ) and a0, . . . , an−1 ∈ A,
{ρ〈a〉 | a ∈ (p)P∗(A)(a0, . . . , an−1)} ⊆ (p)P∗(A/ρ)(ρ〈a0〉, . . . , ρ〈an−1〉)
This inclusion holds if we replace (p)P∗(A) and (p)P∗(A/ρ) by
p ∈ P(n)A (P∗(A)) and p′ ∈ P
(n)A/ρ(P∗(A/ρ)) respectively, where the
polynomial function p′ corresponding to p is given as follows:
(i) if p = cna , then p′ = cnρ〈a〉;
(ii) if p = eni = (xi )P∗(A), then p′ = eni = (xi )P∗(A/ρ);
(iii) if p = fγ(p0, . . . , pnγ−1) and p′0, . . . , p′nγ−1 ∈ P(n)A/ρ(P∗(A/ρ))
are the polynomial functions which correspond to p0, . . . ,
pnγ−1 ∈ P(n)A (P∗(A)) respectively, then
p′ = fγ(p′0, . . . , p′nγ−1).
So, we can write
{ρ〈a〉 | a ∈ p(a0, . . . , an−1)} ⊆ p′(ρ〈a0〉, . . . , ρ〈an−1〉).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Since the canonical map A→ A/ρ is a multialgebrahomomorphism, for any n ∈ N, p ∈ P(n)(τ) and a0, . . . , an−1 ∈ A,
{ρ〈a〉 | a ∈ (p)P∗(A)(a0, . . . , an−1)} ⊆ (p)P∗(A/ρ)(ρ〈a0〉, . . . , ρ〈an−1〉)
This inclusion holds if we replace (p)P∗(A) and (p)P∗(A/ρ) by
p ∈ P(n)A (P∗(A)) and p′ ∈ P
(n)A/ρ(P∗(A/ρ)) respectively, where the
polynomial function p′ corresponding to p is given as follows:
(i) if p = cna , then p′ = cnρ〈a〉;
(ii) if p = eni = (xi )P∗(A), then p′ = eni = (xi )P∗(A/ρ);
(iii) if p = fγ(p0, . . . , pnγ−1) and p′0, . . . , p′nγ−1 ∈ P(n)A/ρ(P∗(A/ρ))
are the polynomial functions which correspond to p0, . . . ,
pnγ−1 ∈ P(n)A (P∗(A)) respectively, then
p′ = fγ(p′0, . . . , p′nγ−1).
So, we can write
{ρ〈a〉 | a ∈ p(a0, . . . , an−1)} ⊆ p′(ρ〈a0〉, . . . , ρ〈an−1〉).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
• Any identity of A is weakly satisfied on A/ρ.
• If A is a universal algebra the multioperations from A/ρ aredefined by the equalities
fγ(ρ〈a0〉, ..., ρ〈anγ−1〉) = {ρ〈b〉|b = fγ(b0, ..., bnγ−1), aiρbi , i = 0, nγ − 1},
and taking an n-ary term p we have
p(ρ〈a0〉, ..., ρ〈an−1〉) ⊇ {ρ〈b〉|b = p(b0, ..., bn−1), aiρbi , i = 0, n − 1}.
In general, this inclusion is not an equality and the identities of Aare only weakly preserved by A/ρ.
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ExampleIf (Z5,+) is the cyclic group of order 5 and we take the equivalence
ρ = {0, 1} × {0, 1} ∪ {2} × {2} ∪ {3, 4} × {3, 4}.
(Z5/ρ,+) is a hypergroupoid on Z5/ρ = {{0, 1}, {2}, {3, 4}} given by
+ {0, 1} {2} {3, 4}{0, 1} {0, 1}, {2} {2}, {3, 4} {0, 1}, {3, 4}{2} {2}, {3, 4} {3, 4} {0, 1}{3, 4} {0, 1}, {3, 4} {0, 1} {0, 1}, {2}, {3, 4}
{ρ〈c〉 | c = (b0 + b1) + b2, b0 = b1 = 2, b2 ∈ {3, 4}} = {ρ〈2〉, ρ〈3〉}
(ρ〈2〉+ ρ〈2〉) + ρ〈3〉 = ρ〈3〉+ ρ〈3〉 = {ρ〈0〉, ρ〈2〉, ρ〈3〉},
and the associativity is only weakly satisfied on (Z5/ρ,+) since
ρ〈2〉+ (ρ〈2〉+ ρ〈3〉) = ρ〈2〉+ ρ〈0〉 = {ρ〈2〉, ρ〈3〉}.
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I Yet, there are identities, like
fγ(x0, . . . , xnγ−1) = fγ(xσ(0), . . . , xσ(nγ−1)) (σ ∈ Snγ ),
which characterize the commutativity of an nγ-ary operationof an algebra, which always hold in a strong manner in thefactor multialgebra.
I QUESTION 2: Can we determine/characterize the identitiesof a (multi)algebra which are always strongly preserved by thefactor multialgebra?
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I Yet, there are identities, like
fγ(x0, . . . , xnγ−1) = fγ(xσ(0), . . . , xσ(nγ−1)) (σ ∈ Snγ ),
which characterize the commutativity of an nγ-ary operationof an algebra, which always hold in a strong manner in thefactor multialgebra.
I QUESTION 2: Can we determine/characterize the identitiesof a (multi)algebra which are always strongly preserved by thefactor multialgebra?
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Example(Roth) If (G , ·) is a finite group and ρ is the conjugacy relation on Gthen G/ρ is the set of the conjugacy classes of G , and the hypergroupoid(G/ρ, ·) is a canonical hypergroup having the identity elementρ〈1〉 = {1}, and for each conjugacy class C from G , the inverse elementis the class C−1 which consists in the inverses of the elements of C . Let1 be the nullary operation which points out the identity element, and −1
the unary operation which associates to each element its inverse. Thenρ〈1〉 and −1 are operations on G/ρ, too, and both multialgebras(G , ·, 1,−1 ) and (G/ρ, ·, ρ〈1〉,−1 ) satisfy the (strong) identities:
(x0 · x1) · x2 = x0 · (x1 · x2), x0 · 1 = 1 · x0 = x0.
QUESTION 3: Can we determine/identify classes of equivalence
relations in (multi)algebra which determine factor multialgebra which
strongly preserves certain (sets of) identities?
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Ideal equivalences
Let A = (A, (fγ)γ<o(τ)) be a multialgebra. An equivalence relationρ on A determines on P∗(A) the relations ρ and ρ defined by:
XρY ⇔ ∀x ∈ X , ∃y ∈ Y : xρy and ∀y ∈ Y , ∃x ∈ X : xρy ;
XρY ⇔ xρy , ∀x ∈ X , ∀y ∈ Y ⇔ X × Y ⊆ ρ.
The relation ρ is an ideal equivalence on A if for any γ < o(τ) andany xi , yi ∈ A for which xiρyi for all i ∈ {0, . . . , nγ − 1} we have
a ∈ fγ(x0, . . . , xnγ−1)⇒ ∃b ∈ fγ(y0, . . . , ynγ−1) : aρb.
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Theorem (Pickett)
If ρ is an ideal equivalence on A then the canonical projection
πρ : A→ A/ρ, πρ(a) = ρ〈a〉
is an ideal homomorphism. Conversely, if h : A→ B is an idealhomomorphism from A into B, its kernel
ker h = {(x , y) ∈ A× A | h(x) = h(y)}
is an ideal equivalence on A. Moreover, the mapping
h(a) 7→ πker h(a)
is an multialgebra isomorphism between h(A) and A/ ker h.
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I (Breaz, Pelea) An equivalence relation ρ of the multialgebraA = (A, (fγ)γ<o(τ)) is ideal if and only if ρ is a congruencerelation on P∗(A).
I The equivalence relations ρ for which the factor multialgebraA/ρ is a universal algebra are particular ideal equivalencerelations.
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I (Breaz, Pelea) An equivalence relation ρ of the multialgebraA = (A, (fγ)γ<o(τ)) is ideal if and only if ρ is a congruencerelation on P∗(A).
I The equivalence relations ρ for which the factor multialgebraA/ρ is a universal algebra are particular ideal equivalencerelations.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Proposition(Breaz, Pelea) Let A = (A, (fγ)γ<o(τ)) be a multialgebra and let ρ be anequivalence relation on A. The following conditions are equivalent:
(a) ρ is an ideal equivalence on A;
(b) for any γ < o(τ), and any xi , yi ∈ A for which xiρyi for alli ∈ {0, . . . , nγ − 1} we have
fγ(x0, . . . , xnγ−1)ρfγ(y0, . . . , ynγ−1);
(c) for any γ < o(τ), any a, b, xi ∈ A (i ∈ {0, . . . , nγ − 1}) such thataρb, and any i ∈ {0, . . . , nγ − 1}, we have
fγ(x0, . . . , xi−1, a, xi+1, . . . , xnγ−1)ρfγ(x0, . . . , xi−1, b, xi+1, . . . , xnγ−1)
(d) for any n ∈ N, any p ∈ P(n)A (P∗(A)), and any xi , yi ∈ A with xiρyi
(i ∈ {0, . . . , n − 1}) we have
p(x0, . . . , xn−1)ρp(y0, . . . , yn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 48: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/48.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Proposition(Breaz, Pelea) Let A = (A, (fγ)γ<o(τ)) be a multialgebra and let ρ be anequivalence relation on A. The following conditions are equivalent:
(a) ρ is an ideal equivalence on A;
(b) for any γ < o(τ), and any xi , yi ∈ A for which xiρyi for alli ∈ {0, . . . , nγ − 1} we have
fγ(x0, . . . , xnγ−1)ρfγ(y0, . . . , ynγ−1);
(c) for any γ < o(τ), any a, b, xi ∈ A (i ∈ {0, . . . , nγ − 1}) such thataρb, and any i ∈ {0, . . . , nγ − 1}, we have
fγ(x0, . . . , xi−1, a, xi+1, . . . , xnγ−1)ρfγ(x0, . . . , xi−1, b, xi+1, . . . , xnγ−1)
(d) for any n ∈ N, any p ∈ P(n)A (P∗(A)), and any xi , yi ∈ A with xiρyi
(i ∈ {0, . . . , n − 1}) we have
p(x0, . . . , xn−1)ρp(y0, . . . , yn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 49: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/49.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Proposition(Breaz, Pelea) Let A = (A, (fγ)γ<o(τ)) be a multialgebra and let ρ be anequivalence relation on A. The following conditions are equivalent:
(a) ρ is an ideal equivalence on A;
(b) for any γ < o(τ), and any xi , yi ∈ A for which xiρyi for alli ∈ {0, . . . , nγ − 1} we have
fγ(x0, . . . , xnγ−1)ρfγ(y0, . . . , ynγ−1);
(c) for any γ < o(τ), any a, b, xi ∈ A (i ∈ {0, . . . , nγ − 1}) such thataρb, and any i ∈ {0, . . . , nγ − 1}, we have
fγ(x0, . . . , xi−1, a, xi+1, . . . , xnγ−1)ρfγ(x0, . . . , xi−1, b, xi+1, . . . , xnγ−1)
(d) for any n ∈ N, any p ∈ P(n)A (P∗(A)), and any xi , yi ∈ A with xiρyi
(i ∈ {0, . . . , n − 1}) we have
p(x0, . . . , xn−1)ρp(y0, . . . , yn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 50: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/50.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Proposition(Breaz, Pelea) Let A = (A, (fγ)γ<o(τ)) be a multialgebra and let ρ be anequivalence relation on A. The following conditions are equivalent:
(a) ρ is an ideal equivalence on A;
(b) for any γ < o(τ), and any xi , yi ∈ A for which xiρyi for alli ∈ {0, . . . , nγ − 1} we have
fγ(x0, . . . , xnγ−1)ρfγ(y0, . . . , ynγ−1);
(c) for any γ < o(τ), any a, b, xi ∈ A (i ∈ {0, . . . , nγ − 1}) such thataρb, and any i ∈ {0, . . . , nγ − 1}, we have
fγ(x0, . . . , xi−1, a, xi+1, . . . , xnγ−1)ρfγ(x0, . . . , xi−1, b, xi+1, . . . , xnγ−1)
(d) for any n ∈ N, any p ∈ P(n)A (P∗(A)), and any xi , yi ∈ A with xiρyi
(i ∈ {0, . . . , n − 1}) we have
p(x0, . . . , xn−1)ρp(y0, . . . , yn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 51: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/51.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Factor multialgebras which are universal algebras
Proposition(Pelea, Purdea) The following conditions are equivalent:
a) A/ρ is a universal algebra;
b) If γ < o(τ), a, b, xi ∈ A, i ∈ {0, . . . , nγ − 1}, with aρb, then
fγ(x0, ..., xi−1, a, xi+1, ..., xnγ−1)ρfγ(x0, ..., xi−1, b, xi+1, ..., xnγ−1)
for all i ∈ {0, . . . , nγ − 1};
c) If γ < o(τ), xi , yi ∈ A and xiρyi for any i ∈ {0, ..., nγ − 1}, then
fγ(x0, . . . , xnγ−1)ρfγ(y0, . . . , ynγ−1);
d) If n ∈ N, p ∈ P(n)A (P∗(A)) and xi , yi ∈ A with xiρyi for any
i ∈ {0, . . . , n − 1}, thenp(x0, . . . , xn−1)ρp(y0, . . . , yn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 52: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/52.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Factor multialgebras which are universal algebras
Proposition(Pelea, Purdea) The following conditions are equivalent:
a) A/ρ is a universal algebra;
b) If γ < o(τ), a, b, xi ∈ A, i ∈ {0, . . . , nγ − 1}, with aρb, then
fγ(x0, ..., xi−1, a, xi+1, ..., xnγ−1)ρfγ(x0, ..., xi−1, b, xi+1, ..., xnγ−1)
for all i ∈ {0, . . . , nγ − 1};
c) If γ < o(τ), xi , yi ∈ A and xiρyi for any i ∈ {0, ..., nγ − 1}, then
fγ(x0, . . . , xnγ−1)ρfγ(y0, . . . , ynγ−1);
d) If n ∈ N, p ∈ P(n)A (P∗(A)) and xi , yi ∈ A with xiρyi for any
i ∈ {0, . . . , n − 1}, thenp(x0, . . . , xn−1)ρp(y0, . . . , yn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 53: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/53.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Factor multialgebras which are universal algebras
Proposition(Pelea, Purdea) The following conditions are equivalent:
a) A/ρ is a universal algebra;
b) If γ < o(τ), a, b, xi ∈ A, i ∈ {0, . . . , nγ − 1}, with aρb, then
fγ(x0, ..., xi−1, a, xi+1, ..., xnγ−1)ρfγ(x0, ..., xi−1, b, xi+1, ..., xnγ−1)
for all i ∈ {0, . . . , nγ − 1};
c) If γ < o(τ), xi , yi ∈ A and xiρyi for any i ∈ {0, ..., nγ − 1}, then
fγ(x0, . . . , xnγ−1)ρfγ(y0, . . . , ynγ−1);
d) If n ∈ N, p ∈ P(n)A (P∗(A)) and xi , yi ∈ A with xiρyi for any
i ∈ {0, . . . , n − 1}, thenp(x0, . . . , xn−1)ρp(y0, . . . , yn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 54: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/54.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Factor multialgebras which are universal algebras
Proposition(Pelea, Purdea) The following conditions are equivalent:
a) A/ρ is a universal algebra;
b) If γ < o(τ), a, b, xi ∈ A, i ∈ {0, . . . , nγ − 1}, with aρb, then
fγ(x0, ..., xi−1, a, xi+1, ..., xnγ−1)ρfγ(x0, ..., xi−1, b, xi+1, ..., xnγ−1)
for all i ∈ {0, . . . , nγ − 1};
c) If γ < o(τ), xi , yi ∈ A and xiρyi for any i ∈ {0, ..., nγ − 1}, then
fγ(x0, . . . , xnγ−1)ρfγ(y0, . . . , ynγ−1);
d) If n ∈ N, p ∈ P(n)A (P∗(A)) and xi , yi ∈ A with xiρyi for any
i ∈ {0, . . . , n − 1}, thenp(x0, . . . , xn−1)ρp(y0, . . . , yn−1).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
We denote by Eua(A) be the set of the relations characterized above.
I Eua(A) is an algebraic closure system on A× A.
⇒ the smallest relation from Eua(A) containing R ⊆ A× A is
α(R) =⋂{ρ ∈ Eua(A) | R ⊆ ρ}.
I α∗ = α(∅) = α(δA) is the smallest element of Eua(A) and it iscalled the fundamental relation of A.
I If an identity q ∩ r 6= ∅ holds in A and ρ ∈ Eua(A) then q = r holdsin A/ρ.
⇒ any identity (weak or strong) which holds on A is also satisfied in itsfundamental algebra A = A/α∗.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
We denote by Eua(A) be the set of the relations characterized above.
I Eua(A) is an algebraic closure system on A× A.
⇒ the smallest relation from Eua(A) containing R ⊆ A× A is
α(R) =⋂{ρ ∈ Eua(A) | R ⊆ ρ}.
I α∗ = α(∅) = α(δA) is the smallest element of Eua(A) and it iscalled the fundamental relation of A.
I If an identity q ∩ r 6= ∅ holds in A and ρ ∈ Eua(A) then q = r holdsin A/ρ.
⇒ any identity (weak or strong) which holds on A is also satisfied in itsfundamental algebra A = A/α∗.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 57: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/57.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
We denote by Eua(A) be the set of the relations characterized above.
I Eua(A) is an algebraic closure system on A× A.
⇒ the smallest relation from Eua(A) containing R ⊆ A× A is
α(R) =⋂{ρ ∈ Eua(A) | R ⊆ ρ}.
I α∗ = α(∅) = α(δA) is the smallest element of Eua(A) and it iscalled the fundamental relation of A.
I If an identity q ∩ r 6= ∅ holds in A and ρ ∈ Eua(A) then q = r holdsin A/ρ.
⇒ any identity (weak or strong) which holds on A is also satisfied in itsfundamental algebra A = A/α∗.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 58: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/58.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
We denote by Eua(A) be the set of the relations characterized above.
I Eua(A) is an algebraic closure system on A× A.
⇒ the smallest relation from Eua(A) containing R ⊆ A× A is
α(R) =⋂{ρ ∈ Eua(A) | R ⊆ ρ}.
I α∗ = α(∅) = α(δA) is the smallest element of Eua(A) and it iscalled the fundamental relation of A.
I If an identity q ∩ r 6= ∅ holds in A and ρ ∈ Eua(A) then q = r holdsin A/ρ.
⇒ any identity (weak or strong) which holds on A is also satisfied in itsfundamental algebra A = A/α∗.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 59: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/59.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
We denote by Eua(A) be the set of the relations characterized above.
I Eua(A) is an algebraic closure system on A× A.
⇒ the smallest relation from Eua(A) containing R ⊆ A× A is
α(R) =⋂{ρ ∈ Eua(A) | R ⊆ ρ}.
I α∗ = α(∅) = α(δA) is the smallest element of Eua(A) and it iscalled the fundamental relation of A.
I If an identity q ∩ r 6= ∅ holds in A and ρ ∈ Eua(A) then q = r holdsin A/ρ.
⇒ any identity (weak or strong) which holds on A is also satisfied in itsfundamental algebra A = A/α∗.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 60: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/60.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
We denote by Eua(A) be the set of the relations characterized above.
I Eua(A) is an algebraic closure system on A× A.
⇒ the smallest relation from Eua(A) containing R ⊆ A× A is
α(R) =⋂{ρ ∈ Eua(A) | R ⊆ ρ}.
I α∗ = α(∅) = α(δA) is the smallest element of Eua(A) and it iscalled the fundamental relation of A.
I If an identity q ∩ r 6= ∅ holds in A and ρ ∈ Eua(A) then q = r holdsin A/ρ.
⇒ any identity (weak or strong) which holds on A is also satisfied in itsfundamental algebra A = A/α∗.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 61: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/61.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I Even if q ∩ r 6= ∅ is not satisfied on A we can obtain a factormultialgebras of A which are universal algebras satisfying q = r byfactorizing it modulo relations from Eua(A) which contain
Rqr =⋃{q(a0, . . . , an−1)× r(a0, . . . , an−1) | a0, ..., an−1 ∈ A}.
I Each relation from Eua(A) which gives a factor multialgebrasatisfying the identity q = r must contain the relation Rqr.
⇒ the smallest relation from Eua(A) for which the factor multialgebrais a universal algebra satisfying q = r is α∗qr = α(Rqr).
(In particular, α∗ = α∗x0x0.)
I (Pelea, Purdea) α∗qr is the transitive closure of the relation αqr
defined by
xαqry ⇔ ∃p ∈ P(1)A (P∗(A)), ∃ a0, . . . , an−1 ∈ A :
x ∈ p(q(a0, . . . , an−1)), y ∈ p(r(a0, . . . , an−1)) or
y ∈ p(q(a0, . . . , an−1)), x ∈ p(r(a0, . . . , an−1)).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 62: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/62.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I Even if q ∩ r 6= ∅ is not satisfied on A we can obtain a factormultialgebras of A which are universal algebras satisfying q = r byfactorizing it modulo relations from Eua(A) which contain
Rqr =⋃{q(a0, . . . , an−1)× r(a0, . . . , an−1) | a0, ..., an−1 ∈ A}.
I Each relation from Eua(A) which gives a factor multialgebrasatisfying the identity q = r must contain the relation Rqr.
⇒ the smallest relation from Eua(A) for which the factor multialgebrais a universal algebra satisfying q = r is α∗qr = α(Rqr).
(In particular, α∗ = α∗x0x0.)
I (Pelea, Purdea) α∗qr is the transitive closure of the relation αqr
defined by
xαqry ⇔ ∃p ∈ P(1)A (P∗(A)), ∃ a0, . . . , an−1 ∈ A :
x ∈ p(q(a0, . . . , an−1)), y ∈ p(r(a0, . . . , an−1)) or
y ∈ p(q(a0, . . . , an−1)), x ∈ p(r(a0, . . . , an−1)).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 63: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/63.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I Even if q ∩ r 6= ∅ is not satisfied on A we can obtain a factormultialgebras of A which are universal algebras satisfying q = r byfactorizing it modulo relations from Eua(A) which contain
Rqr =⋃{q(a0, . . . , an−1)× r(a0, . . . , an−1) | a0, ..., an−1 ∈ A}.
I Each relation from Eua(A) which gives a factor multialgebrasatisfying the identity q = r must contain the relation Rqr.
⇒ the smallest relation from Eua(A) for which the factor multialgebrais a universal algebra satisfying q = r is α∗qr = α(Rqr).
(In particular, α∗ = α∗x0x0.)
I (Pelea, Purdea) α∗qr is the transitive closure of the relation αqr
defined by
xαqry ⇔ ∃p ∈ P(1)A (P∗(A)), ∃ a0, . . . , an−1 ∈ A :
x ∈ p(q(a0, . . . , an−1)), y ∈ p(r(a0, . . . , an−1)) or
y ∈ p(q(a0, . . . , an−1)), x ∈ p(r(a0, . . . , an−1)).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 64: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/64.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I Even if q ∩ r 6= ∅ is not satisfied on A we can obtain a factormultialgebras of A which are universal algebras satisfying q = r byfactorizing it modulo relations from Eua(A) which contain
Rqr =⋃{q(a0, . . . , an−1)× r(a0, . . . , an−1) | a0, ..., an−1 ∈ A}.
I Each relation from Eua(A) which gives a factor multialgebrasatisfying the identity q = r must contain the relation Rqr.
⇒ the smallest relation from Eua(A) for which the factor multialgebrais a universal algebra satisfying q = r is α∗qr = α(Rqr).
(In particular, α∗ = α∗x0x0.)
I (Pelea, Purdea) α∗qr is the transitive closure of the relation αqr
defined by
xαqry ⇔ ∃p ∈ P(1)A (P∗(A)), ∃ a0, . . . , an−1 ∈ A :
x ∈ p(q(a0, . . . , an−1)), y ∈ p(r(a0, . . . , an−1)) or
y ∈ p(q(a0, . . . , an−1)), x ∈ p(r(a0, . . . , an−1)).
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
![Page 65: Term functions and polynomial functions in hyperstructure theorymath.ubbcluj.ro/.../conferinte/PurdeaPelea_Iasi11.pdf · 2014-03-20 · Open problems in algebraic hyperstructure theory](https://reader035.fdocuments.net/reader035/viewer/2022062921/5f03a2717e708231d40a050a/html5/thumbnails/65.jpg)
Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
I Even if q ∩ r 6= ∅ is not satisfied on A we can obtain a factormultialgebras of A which are universal algebras satisfying q = r byfactorizing it modulo relations from Eua(A) which contain
Rqr =⋃{q(a0, . . . , an−1)× r(a0, . . . , an−1) | a0, ..., an−1 ∈ A}.
I Each relation from Eua(A) which gives a factor multialgebrasatisfying the identity q = r must contain the relation Rqr.
⇒ the smallest relation from Eua(A) for which the factor multialgebrais a universal algebra satisfying q = r is α∗qr = α(Rqr).
(In particular, α∗ = α∗x0x0.)
I (Pelea, Purdea) α∗qr is the transitive closure of the relation αqr
defined by
xαqry ⇔ ∃p ∈ P(1)A (P∗(A)), ∃ a0, . . . , an−1 ∈ A :
x ∈ p(q(a0, . . . , an−1)), y ∈ p(r(a0, . . . , an−1)) or
y ∈ p(q(a0, . . . , an−1)), x ∈ p(r(a0, . . . , an−1)).
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The fundamental relation of a multialgebra
(Pelea) α∗ is the transitive closure of the relation α defined by
xαy ⇔ ∃n ∈ N, ∃p ∈ P(n)A (P∗(A)), ∃a0, . . . , an−1 ∈ A :
x , y ∈ p(a0, . . . , an−1);
I For any polynomial function p ∈ P(n)A (P∗(A)) and any
a0, . . . , an−1 ∈ A, there exist m ∈ N, m ≥ n, b0, . . . , bm−1 ∈ A anda term function p′ ∈ P(m)(P∗(A)) such that
p(a0, . . . , an−1) = p′(b0, . . . , bm−1).
⇒ we can redefine α as follows
xαy ⇔ ∃n ∈ N, ∃p ∈ P(n)(P∗(A)), ∃a0, . . . , an−1 ∈ A :
x , y ∈ p(a0, . . . , an−1)
I (Pelea, Purdea) α∗ is also the transitive closure of the relation
xα′y ⇔ ∃p ∈ P(1)A (P∗(A)), ∃a ∈ A : x , y ∈ p(a).
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The fundamental relation of a multialgebra
(Pelea) α∗ is the transitive closure of the relation α defined by
xαy ⇔ ∃n ∈ N, ∃p ∈ P(n)A (P∗(A)), ∃a0, . . . , an−1 ∈ A :
x , y ∈ p(a0, . . . , an−1);
I For any polynomial function p ∈ P(n)A (P∗(A)) and any
a0, . . . , an−1 ∈ A, there exist m ∈ N, m ≥ n, b0, . . . , bm−1 ∈ A anda term function p′ ∈ P(m)(P∗(A)) such that
p(a0, . . . , an−1) = p′(b0, . . . , bm−1).
⇒ we can redefine α as follows
xαy ⇔ ∃n ∈ N, ∃p ∈ P(n)(P∗(A)), ∃a0, . . . , an−1 ∈ A :
x , y ∈ p(a0, . . . , an−1)
I (Pelea, Purdea) α∗ is also the transitive closure of the relation
xα′y ⇔ ∃p ∈ P(1)A (P∗(A)), ∃a ∈ A : x , y ∈ p(a).
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The fundamental relation of a multialgebra
(Pelea) α∗ is the transitive closure of the relation α defined by
xαy ⇔ ∃n ∈ N, ∃p ∈ P(n)A (P∗(A)), ∃a0, . . . , an−1 ∈ A :
x , y ∈ p(a0, . . . , an−1);
I For any polynomial function p ∈ P(n)A (P∗(A)) and any
a0, . . . , an−1 ∈ A, there exist m ∈ N, m ≥ n, b0, . . . , bm−1 ∈ A anda term function p′ ∈ P(m)(P∗(A)) such that
p(a0, . . . , an−1) = p′(b0, . . . , bm−1).
⇒ we can redefine α as follows
xαy ⇔ ∃n ∈ N, ∃p ∈ P(n)(P∗(A)), ∃a0, . . . , an−1 ∈ A :
x , y ∈ p(a0, . . . , an−1)
I (Pelea, Purdea) α∗ is also the transitive closure of the relation
xα′y ⇔ ∃p ∈ P(1)A (P∗(A)), ∃a ∈ A : x , y ∈ p(a).
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
The fundamental relation of a multialgebra
(Pelea) α∗ is the transitive closure of the relation α defined by
xαy ⇔ ∃n ∈ N, ∃p ∈ P(n)A (P∗(A)), ∃a0, . . . , an−1 ∈ A :
x , y ∈ p(a0, . . . , an−1);
I For any polynomial function p ∈ P(n)A (P∗(A)) and any
a0, . . . , an−1 ∈ A, there exist m ∈ N, m ≥ n, b0, . . . , bm−1 ∈ A anda term function p′ ∈ P(m)(P∗(A)) such that
p(a0, . . . , an−1) = p′(b0, . . . , bm−1).
⇒ we can redefine α as follows
xαy ⇔ ∃n ∈ N, ∃p ∈ P(n)(P∗(A)), ∃a0, . . . , an−1 ∈ A :
x , y ∈ p(a0, . . . , an−1)
I (Pelea, Purdea) α∗ is also the transitive closure of the relation
xα′y ⇔ ∃p ∈ P(1)A (P∗(A)), ∃a ∈ A : x , y ∈ p(a).
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Let B be a universal algebra and ρ an equivalence relation on B. Denoteby θ(ρ) the smallest congruence relation on B containing ρ.
I (Pelea, Purdea) For n ∈ N, p ∈ P(n)B/ρ(P∗(B/ρ)) and
x , y , z0, . . . , zn−1 ∈ B we have
ρ〈x〉, ρ〈y〉 ∈ p(ρ〈z0〉, . . . , ρ〈zn−1〉) ⇒ xθ(ρ)y .
⇒ (Pelea, Purdea)B/ρ ∼= B/θ(ρ).
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Let B be a universal algebra and ρ an equivalence relation on B. Denoteby θ(ρ) the smallest congruence relation on B containing ρ.
I (Pelea, Purdea) For n ∈ N, p ∈ P(n)B/ρ(P∗(B/ρ)) and
x , y , z0, . . . , zn−1 ∈ B we have
ρ〈x〉, ρ〈y〉 ∈ p(ρ〈z0〉, . . . , ρ〈zn−1〉) ⇒ xθ(ρ)y .
⇒ (Pelea, Purdea)B/ρ ∼= B/θ(ρ).
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Example
Let (G , ·) be a group, H a subgroup of G ,
G/H = {xH | x ∈ G}
and(xH)(yH) = {zH | z = x ′y ′, x ′ ∈ xH, y ′ ∈ yH}.
The hypergroupoid (G/H, ·) is a hypergroup (Marty).
If G/H is the fundamental group of the hypergroup G/H and H isthe smallest normal subgroup of G which contains H then
G/H ∼= G/H.
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Direct products of multialgebras
Let (Ai | i ∈ I ) be a family of multialgebras of type τ . TheCartesian product
∏i∈I Ai with the multioperations
fγ((a0i )i∈I , . . . , (a
nγ−1i )i∈I ) =
∏i∈I
fγ(a0i , . . . , a
nγ−1i ),
is a multialgebra called the direct product of the multialgebras(Ai | i ∈ I ).
I If p ∈ P(n)(τ) and (a0i )i∈I , . . . , (a
n−1i )i∈I ∈
∏i∈I Ai then
p((a0i )i∈I , . . . , (a
n−1i )i∈I ) =
∏i∈I
p(a0i , . . . , a
n−1i ).
I The direct product of a family of multialgebras which satisfy acertain identity (weak or strong) satisfies the same identity.
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Direct products of multialgebras
Let (Ai | i ∈ I ) be a family of multialgebras of type τ . TheCartesian product
∏i∈I Ai with the multioperations
fγ((a0i )i∈I , . . . , (a
nγ−1i )i∈I ) =
∏i∈I
fγ(a0i , . . . , a
nγ−1i ),
is a multialgebra called the direct product of the multialgebras(Ai | i ∈ I ).
I If p ∈ P(n)(τ) and (a0i )i∈I , . . . , (a
n−1i )i∈I ∈
∏i∈I Ai then
p((a0i )i∈I , . . . , (a
n−1i )i∈I ) =
∏i∈I
p(a0i , . . . , a
n−1i ).
I The direct product of a family of multialgebras which satisfy acertain identity (weak or strong) satisfies the same identity.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Direct products of multialgebras
Let (Ai | i ∈ I ) be a family of multialgebras of type τ . TheCartesian product
∏i∈I Ai with the multioperations
fγ((a0i )i∈I , . . . , (a
nγ−1i )i∈I ) =
∏i∈I
fγ(a0i , . . . , a
nγ−1i ),
is a multialgebra called the direct product of the multialgebras(Ai | i ∈ I ).
I If p ∈ P(n)(τ) and (a0i )i∈I , . . . , (a
n−1i )i∈I ∈
∏i∈I Ai then
p((a0i )i∈I , . . . , (a
n−1i )i∈I ) =
∏i∈I
p(a0i , . . . , a
n−1i ).
I The direct product of a family of multialgebras which satisfy acertain identity (weak or strong) satisfies the same identity.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory
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Direct limits of direct systems of multialgebras
Let A = ((Ai | i ∈ I ), (ϕij | i , j ∈ I , i ≤ j)) be a direct system ofmultialgebras and let A∞ be the direct limit of the direct system oftheir supporting sets.
Let us remind that:
I (I ,≤) is a directed preordered set;
I the set A∞ is the factor of the disjoint union A of the sets Ai
modulo the equivalence relation ≡ defined as follows: for anyx , y ∈ A there exist i , j ∈ I , such that x ∈ Ai , y ∈ Aj , and
x ≡ y ⇔ ∃k ∈ I , i ≤ k, j ≤ k : ϕik(x) = ϕjk(y).
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Direct limits of direct systems of multialgebras
Let A = ((Ai | i ∈ I ), (ϕij | i , j ∈ I , i ≤ j)) be a direct system ofmultialgebras and let A∞ be the direct limit of the direct system oftheir supporting sets.
Let us remind that:
I (I ,≤) is a directed preordered set;
I the set A∞ is the factor of the disjoint union A of the sets Ai
modulo the equivalence relation ≡ defined as follows: for anyx , y ∈ A there exist i , j ∈ I , such that x ∈ Ai , y ∈ Aj , and
x ≡ y ⇔ ∃k ∈ I , i ≤ k, j ≤ k : ϕik(x) = ϕjk(y).
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Open problems in algebraic hyperstructure theory and applications, September 24–26, 2011, Iasi, Romania
Direct limits of direct systems of multialgebras
Let A = ((Ai | i ∈ I ), (ϕij | i , j ∈ I , i ≤ j)) be a direct system ofmultialgebras and let A∞ be the direct limit of the direct system oftheir supporting sets.
Let us remind that:
I (I ,≤) is a directed preordered set;
I the set A∞ is the factor of the disjoint union A of the sets Ai
modulo the equivalence relation ≡ defined as follows: for anyx , y ∈ A there exist i , j ∈ I , such that x ∈ Ai , y ∈ Aj , and
x ≡ y ⇔ ∃k ∈ I , i ≤ k, j ≤ k : ϕik(x) = ϕjk(y).
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We define the multioperations fγ on A∞ = {x | x ∈ A} as follows:if x0, . . . , xnγ−1 ∈ A∞ and for any j ∈ {0, ..., nγ − 1} we considerthat xj ∈ Aij (ij ∈ I ) then
fγ(a0, . . . , anγ−1) = {a ∈ A∞ | ∃m ∈ I , i0 ≤ m, . . . , inγ−1 ≤ m,
a ∈ fγ(ϕi0m(a0), . . . , ϕinγ−1m(anγ−1))}.
The multialgebra A∞ = (A∞, (fγ)γ<o(τ)) is called the direct limitof the direct system A.
I If p ∈ P(n)(τ), a0, . . . , an−1 ∈ A and i0, . . . , in−1 ∈ I are suchthat aj ∈ Aij for all j ∈ {0, . . . , n − 1} then
p(a0, . . . , an−1) = {a ∈ A∞ | ∃m ∈ I , i0 ≤ m, . . . , in−1 ≤ m,
a ∈ p(ϕi0m(a0), . . . , ϕin−1m(an−1))}.
I The direct limit of a direct system of multialgebras whichsatisfy a certain identity (weak or strong) satisfies the sameidentity.
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We define the multioperations fγ on A∞ = {x | x ∈ A} as follows:if x0, . . . , xnγ−1 ∈ A∞ and for any j ∈ {0, ..., nγ − 1} we considerthat xj ∈ Aij (ij ∈ I ) then
fγ(a0, . . . , anγ−1) = {a ∈ A∞ | ∃m ∈ I , i0 ≤ m, . . . , inγ−1 ≤ m,
a ∈ fγ(ϕi0m(a0), . . . , ϕinγ−1m(anγ−1))}.
The multialgebra A∞ = (A∞, (fγ)γ<o(τ)) is called the direct limitof the direct system A.
I If p ∈ P(n)(τ), a0, . . . , an−1 ∈ A and i0, . . . , in−1 ∈ I are suchthat aj ∈ Aij for all j ∈ {0, . . . , n − 1} then
p(a0, . . . , an−1) = {a ∈ A∞ | ∃m ∈ I , i0 ≤ m, . . . , in−1 ≤ m,
a ∈ p(ϕi0m(a0), . . . , ϕin−1m(an−1))}.
I The direct limit of a direct system of multialgebras whichsatisfy a certain identity (weak or strong) satisfies the sameidentity.
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We define the multioperations fγ on A∞ = {x | x ∈ A} as follows:if x0, . . . , xnγ−1 ∈ A∞ and for any j ∈ {0, ..., nγ − 1} we considerthat xj ∈ Aij (ij ∈ I ) then
fγ(a0, . . . , anγ−1) = {a ∈ A∞ | ∃m ∈ I , i0 ≤ m, . . . , inγ−1 ≤ m,
a ∈ fγ(ϕi0m(a0), . . . , ϕinγ−1m(anγ−1))}.
The multialgebra A∞ = (A∞, (fγ)γ<o(τ)) is called the direct limitof the direct system A.
I If p ∈ P(n)(τ), a0, . . . , an−1 ∈ A and i0, . . . , in−1 ∈ I are suchthat aj ∈ Aij for all j ∈ {0, . . . , n − 1} then
p(a0, . . . , an−1) = {a ∈ A∞ | ∃m ∈ I , i0 ≤ m, . . . , in−1 ≤ m,
a ∈ p(ϕi0m(a0), . . . , ϕin−1m(an−1))}.
I The direct limit of a direct system of multialgebras whichsatisfy a certain identity (weak or strong) satisfies the sameidentity.
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For details, see
Breaz, S.; Pelea, C., Multialgebras and term functions over the algebra of their nonvoid subsets,
Mathematica (Cluj), 43(66), 2, 2001, 143–149.
Pelea, C.: On the fundamental relation of a multialgebra, Ital. J. Pure Appl. Math., 10, 2001, 141–146.
Pelea, C.: On the direct product of multialgebras, Studia Univ. Babes–Bolyai Math., 48, 2, 2003, 93–98.
Pelea, C.: On the direct limit of a direct family of multialgebras, Discrete Mathematics, 306, 22, 2006,
2916–2930.
Pelea, C.; Purdea, I.: Multialgebras, universal algebras and identities, J. Aust. Math. Soc, 81, 2006,
121–139.
Pelea, C.; Purdea, I., Identities in multialgebra theory, Proceedings of the 10th International Congress on
Algebraic Hyperstructures and Applications, Brno 2008, 2009, 251–266.
Ioan Purdea, Cosmin Pelea Term functions and polynomial functions in hyperstructure theory