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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Axiomatization of quasigroups

100%
Smith J.
Discussiones Mathematicae - General Algebra and Applications
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2007
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tom 27
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nr 1
EN
Quasigroups were originally described combinatorially, in terms of existence and uniqueness conditions on the solutions to certain equations. Evans introduced a universal-algebraic characterization, as algebras with three binary operations satisfying four identities. Now, quasigroups are redefined as heterogeneous algebras, satisfying just two conditions respectively known as hypercommutativity and hypercancellativity.
2
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Hyperidentities in many-sorted algebras

100%
Denecke K., Lekkoksung S.
Discussiones Mathematicae - General Algebra and Applications
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2009
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tom 29
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nr 1
47-74
EN
The theory of hyperidentities generalizes the equational theory of universal algebras and is applicable in several fields of science, especially in computers sciences (see e.g. [2,1]). The main tool to study hyperidentities is the concept of a hypersubstitution. Hypersubstitutions of many-sorted algebras were studied in [3]. On the basis of hypersubstitutions one defines a pair of closure operators which turns out to be a conjugate pair. The theory of conjugate pairs of additive closure operators can be applied to characterize solid varieties, i.e., varieties in which every identity is satisfied as a hyperidentity (see [4]). The aim of this paper is to apply the theory of conjugate pairs of additive closure operators to many-sorted algebras.
3
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Hyperidentities in transitive graph algebras

75%
Poomsa-ard T., Wetweerapong J., Samartkoon C.
Discussiones Mathematicae - General Algebra and Applications
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2005
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tom 25
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nr 1
23-37
EN
Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G = (V,E) is called a transitive graph if the corresponding graph algebra A(G) satisfies the equation x(yz) ≈ (xz)(yz). An identity s ≈ t of terms s and t of any type t is called a hyperidentity of an algebra A̲ if whenever the operation symbols occurring in s and t are replaced by any term operations of A of the appropriate arity, the resulting identities hold in A̲ . In this paper we characterize transitive graph algebras, identities and hyperidentities in transitive graph algebras.
4
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Special m-hyperidentities in biregular leftmost graph varieties of type (2,0)

63%
Anantpinitwatna A., Poomsa-ard T.
Discussiones Mathematicae - General Algebra and Applications
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2009
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tom 29
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nr 2
81-107
EN
Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra $\underline{A(G)}$ satisfies s ≈ t. A class of graph algebras V is called a graph variety if $V = Mod_g Σ$ where Σ is a subset of T(X) × T(X). A graph variety $V' = Mod_gΣ'$ is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if $\underline{A(G)}$ satisfies s ≈ t for all G ∈ V. An identity s ≈ t of a variety V is called a hyperidentity of a graph algebra $\underline{A(G)}$, G ∈ V whenever the operation symbols occuring in s and t are replaced by any term operations of $\underline{A(G)}$ of the appropriate arity, the resulting identities hold in $\underline{A(G)}$. An identity s ≈ t of a variety V is called an M-hyperidentity of a graph algebra $\underline{A(G)}$, G ∈ V whenever the operation symbols occuring in s and t are replaced by any term operations in a subgroupoid M of term operations of $\underline{A(G)}$ of the appropriate arity, the resulting identities hold in $\underline{A(G)}$. In this paper we characterize special M-hyperidentities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [3].
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