McCammond’s normal forms for free aperiodic semigroups revisited

A new scheme for proving pseudoidentities from a given set Σ of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when Σ defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when Σ defines a pseudovariety V which is σ-reducible for the equation x = y, provided Σ is enough to prove a basis of identities for the variety of σ-algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.

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“…The following describes a proof from Σ. First, we do an algebraic proof: (7) (xy) ω = (xy) ω+1 = x(yx) ω y = x(xy) ω y = · · · = x m (xy) ω y m .…”

Section: 2mentioning

confidence: 99%

Towards a pseudoequational proof theory

Almeida

Klíma

2018

Port. Math.

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“…A Lyndon word is a primitive word which is minimal, with respect to the lexicographic ordering, in its conjugacy class. For combinatorial properties involving Lyndon words that are relevant for the remaining of this paper, the reader is referred to [3], where an alternative proof of correctness of McCammond's normal form algorithm over A is presented. We employ the following notation for κ-terms, where n > 0: x ω+n represents x ω x n ; x ω+0 is x ω ; x ω−n denotes (x ω−1 ) n .…”

Section: Canonical Forms For κ-Terms Of Rankmentioning

confidence: 99%

Semigroup presentations for test local groups

2014

Self Cite

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In this paper we exhibit a type of semigroup presentations which determines a class of local groups. We show that the finite elements of this class generate the pseudovariety LG of all finite local groups and use them as test-semigroups to prove that LG and S, the pseudovariety of all finite semigroups, verify the same κ-identities involving κ-terms of rank at most 1, where κ denotes the implicit signature consisting of the multiplication and the (ω − 1)-power. of all finite semigroups S such that eSe ∈ H for each idempotent e of S, and we recall thatwhere D is the pseudovariety of all finite semigroups whose idempotents are right zeros. It is well known (see [13] for a proof) that a finite semigroup S is a local group if and only if all the idempotents of S lie in the minimal ideal of S. A proof of the generalization of this result to arbitrary semigroups can be found in Proposition 2.1 below. There, a semigroup S is characterized as being a local group if and only if S has no idempotents or S has a minimal ideal J which is a completely simple semigroup that contains all the idempotents of S. In this case, by the Rees-Suschkewitsch Theorem, J is isomorphic to a Rees matrix semigroup over a group (the maximal subgroup of S). In [9], Howie and Ruškuc showed how to find a semigroup presentation for a Rees matrix semigroup M[G; I, Λ; P ] given a semigroup presentation for the group G.In [2] (see also [1, Section 10.6]), Almeida and Azevedo showed that a semidirect product of the form V * D, with the pseudovariety V not locally trivial, is generated by a class formed by certain semigroups M k (S, ) with k ≥ 1, S ∈ V, A an alphabet and : A + → S a ksuperposition homomorphism. Therefore, possible properties of V * D may be tested on the semigroups M k (S, ) and Almeida and Azevedo applied those test-semigroups (an expression used in [1]) to obtain a representation of the free pro-(V * D) semigroup over A.

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“…The importance of profinite monoids in automata theory and finite semigroup theory was first highlighted, starting in the late eighties, by Almeida [1]; also see the more recent monograph by Rhodes and the second-named author [30], or Straubing and Weil's handbook article [36]. The structure of free pro-aperiodic monoids has been studied recently by several authors, e.g., [4,20,22,6], but many difficult questions remain open. Many of the existing results about free pro-aperiodic monoids are about the submonoid of elements definable by ω-terms and rely on an ingenuous normal form algorithm due to…”

Section: Introductionmentioning

confidence: 99%

“…The separation of normal forms makes use of his solution to the word problem for free Burnside semigroups of sufficiently large exponent [25], which inspired the definition of the normal forms in the first place. Recently, Almeida, Costa and Zeitoun [6] [22] a new algorithm to solve the word problem for ω-terms in F A (A) using model-theoretic ideas. They assign the same A-word that we did above to each ω-term.…”

mentioning

confidence: 99%

Pro-aperiodic monoids via saturated models

Gool

Steinberg

2019

Isr. J. Math.

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We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids. Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for ω-terms that avoids using McCammond's normal forms, as well as new proofs and extensions of other structural results concerning free pro-aperiodic monoids.

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McCammond’s normal forms for free aperiodic semigroups revisited

Cited by 6 publications

References 20 publications

Towards a pseudoequational proof theory

Towards a pseudoequational proof theory

Semigroup presentations for test local groups

Pro-aperiodic monoids via saturated models

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