Functional Menger   -Algebras

Abstract. We consider the identities of a variety of semigroup-related algebras modeling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable.

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“…This proposition also follows from the n = 1 case of a more general result for multiplace functions obtained by Dudek and Trokhimenko [4].…”

Section: Proposition 22 [12]supporting

confidence: 69%

Identities in the Algebra of Partial Maps

Jackson

Stokes

2006

Int. J. Algebra Comput.

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“…Algebras of the form ∩ 1 n , where ∩ is a set-theoretic intersection, are called functional Menger ∩-algebras of n-place functions. In the literature, such algebras are also called functional Menger -algebras (see Dudek and Trokhimenko, 2002;Trokhimenko, 2000).…”

Section: Preliminariesmentioning

confidence: 99%

“…Functional Menger -algebras and Menger systems of rank n are isomorphic, respectively, to some functional Menger ∩-algebras and Menger systems of n-place functions (see Dudek and Trokhimenko, 2002;Trokhimenko, 1973). Each homomorphism of such abstract algebras into corresponding algebras of n-place functions is called a representation by n-place functions.…”

Section: Preliminariesmentioning

confidence: 99%

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Stabilizers of Functional Menger Systems

Dudek¹,

Trokhimenko²

2009

Communications in Algebra

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“…The subalgebras of X ∧ are characterised by Garvac'kiȋ [9]. Subalgebras of X ∧ R are characterised by Dudek and Trokhimenko [6] and the authors [14]. The class is a variety, and its equational theory is algorithmically described by the authors [15].…”

Section: Introductionmentioning

confidence: 99%

Partial Maps with Domain and Range: Extending Schein's Representation

Jackson

Stokes

2009

Communications in Algebra

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The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein [25] gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced-as type A semigroups-by Fountain, [7]). We provide an account of Schein's result (which until now appears only in Russian) and extend Schein's method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised.

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Functional Menger -Algebras

Cited by 19 publications

References 4 publications

Identities in the Algebra of Partial Maps

Identities in the Algebra of Partial Maps

Stabilizers of Functional Menger Systems

Partial Maps with Domain and Range: Extending Schein's Representation

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