Identities in the Algebra of Partial Maps

Jackson, Marcel; Stokes, Tim

doi:10.1142/s0218196706003426

Cited by 11 publications

(10 citation statements)

References 19 publications

(42 reference statements)

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“…We conjecture that the equational problem has a similar solution to that presented for B-semigroups in Section 5. (An analogous case for partial functions with domain and * has such a solution: see [8].) But we observe that the non-finite basis argument given in Section 6 will not carry over to the agreeable case.…”

Section: Proofmentioning

confidence: 87%

SEMIGROUPS WITH if–then–else AND HALTING PROGRAMS

Jackson

Stokes

2009

Int. J. Algebra Comput.

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The "if–then–else" construction is one of the most elementary programming commands, and its abstract laws have been widely studied, starting with McCarthy. Possibly, the most obvious extension of this is to include the operation of composition of programs, which gives a semigroup of functions (total, partial, or possibly general binary relations) that can be recombined using if–then–else. We show that this particular extension admits no finite complete axiomatization and instead focus on the case where composition of functions with predicates is also allowed (and we argue there is good reason to take this approach). In the case of total functions — modeling halting programs — we give a complete axiomatization for the theory in terms of a finite system of equations. We obtain a similar result when an operation of equality test and/or fixed point test is included.

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Section: Proofmentioning

confidence: 87%

SEMIGROUPS WITH if–then–else AND HALTING PROGRAMS

Jackson

Stokes

2009

Int. J. Algebra Comput.

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“…While this connection is weak and imprecise, it is this interpretation that provides the central idea underlying the algorithmic description of the equational theory of representable SLORCs in Jackson and Stokes [15].…”

Section: Preliminaries: a First Approximationmentioning

confidence: 99%

“…The equational theory of twisted agreeable semigroups with or without 0 and 1 is effectively described in Jackson and Stokes [15] (and hence an effective algorithm is known for the equational theory of representable -semigroups with ≤ * 0 1 ). We now investigate the representable R I -semigroups with or without 0 or 1.…”

Section: Domain Operationsmentioning

confidence: 99%

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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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“…In the same way, the reader will see that the class of all semigroups representable as a system of permutations on a set (let us call this G), is the same as the semigroup subreducts of the variety of groups, or equivalently the class of semigroups embeddable in a group. As these examples demonstrate, quasivarieties arise frequently in the study of representations of algebraic structures by way of particular kinds of maps (in the cases above, as injective partial maps and by permutations on a set respectively); see [29] and [10] for surveys of this rich area.…”

Section: Background and Motivationmentioning

confidence: 99%

Relatively inherently nonfinitely q-based semigroups

Jackson¹,

Volkov²

2008

Trans. Amer. Math. Soc.

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Abstract. We prove that every semigroup S whose quasivariety contains a 3-nilpotent semigroup or a semigroup of index more than 2 has no finite basis for its quasi-identities provided that one of the following properties holds:• S is finite;• S has a faithful representation by injective partial maps on a set;• S has a faithful representation by order preserving maps on a chain. As a corollary it is shown that, in an asymptotic sense, almost all finite semigroups and finite monoids admit no finite basis for their quasi-identities. Background and motivation

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Identities in the Algebra of Partial Maps

Cited by 11 publications

References 19 publications

SEMIGROUPS WITH if–then–else AND HALTING PROGRAMS

SEMIGROUPS WITH if–then–else AND HALTING PROGRAMS

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

Relatively inherently nonfinitely q-based semigroups

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