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EFree Objects and ELocality for Completely Regular Semigroups
, 1996
"... We prove that the evariety CR(H), of all completely regular semigroups whose subgroups belong to some group variety H , is elocal; that is, every regular, locally completely regular semigroupoid [with subgroups from H ] divides a completely regular semigroup [with subgroups from H ], in a `reg ..."
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We prove that the evariety CR(H), of all completely regular semigroups whose subgroups belong to some group variety H , is elocal; that is, every regular, locally completely regular semigroupoid [with subgroups from H ] divides a completely regular semigroup [with subgroups from H ], in a `regular' way. In a future paper with P.G. Trotter, this theorem will be applied to semidirect products of evarieties and to efree Esolid regular semigroups. A key role in the proof is played by the efree semigroups in the evariety CR(H) . We provide a solution to the `word problem' in these semigroups, in the style of that for free completely regular semigroups given by Ka dourek and Pol`ak. The solution is derived from the author's work on free products of completely regular semigroups. In an earlier paper [3] the author proved that any category whose local (or `loop') monoids satisfy the identity x n+1 = x divides a monoid satisfying the same identity. This result is easily tra...
Algorithmic Problems for Finite Groups and Finite 0Simple Semigroups
, 1996
"... It is shown that the embeddability of a finite 4nilpotent semigroup into a 0simple finite semigroup with maximal groups from a pseudovariety V is decidable if and only if the universal theory of the class V is decidable. We show that it is impossible to replace 4 by 3 in this statement. We also sho ..."
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It is shown that the embeddability of a finite 4nilpotent semigroup into a 0simple finite semigroup with maximal groups from a pseudovariety V is decidable if and only if the universal theory of the class V is decidable. We show that it is impossible to replace 4 by 3 in this statement. We also show that if the membership in V is decidable then the membership in the pseudovariety generated by the class of all finite 0simple semigroups with subgroups from V is decidable while the membership in the quasivariety generated by this class of 0simple semigroups may be undecidable. 1 Introduction One of the most important classes of semigroups is the class of 0simple finite semigroups. Recall that a semigroup is called 0simple if it does not have ideals except itself and possibly f0g. Every finite semigroup may be obtained from 0simple semigroups by a sequence of ideal extensions. The classic theorem of Sushkevich [3] (which was arguably the first theorem in the algebraic theory of sem...
Rees Matrix Covers and Semidirect Products of Regular Semigroups
, 1996
"... In a recent paper, P.G. Trotter and the author introduced a "regular" semidirect product UV of evarieties U and V. Among several specific situations investigated there was the case V = RZ, the evariety of right zero semigroups. Applying a covering theorem of McAlister, it was shown th ..."
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In a recent paper, P.G. Trotter and the author introduced a "regular" semidirect product UV of evarieties U and V. Among several specific situations investigated there was the case V = RZ, the evariety of right zero semigroups. Applying a covering theorem of McAlister, it was shown there that in several important cases (for instance for the evariety of inverse semigroups), U RZ is precisely the evariety LU of "locally  U" semigroups. The main result of the current paper characterizes membership of a regular semigroup S in U RZ in a number of ways, one in terms of an associated category SE and another in terms of S regularly dividing a regular Rees matrix semigroup over a member of U. The categorical condition leads directly to a characterization of the equality U RZ = LU in terms of a graphical condition on U, slightly weaker than `elocality'. Among consequences of known results on elocality, the conjecture CR RZ = LCR, (with CR denoting the evariety of complete...
Decidable and undecidable problems related to completely 0simple semigroups
, 1996
"... The undecidable problems of the title are concerned with the question: is a given finite semigroup embeddable in a given type of completely 0simple semigroups? It is shown, for example, that the embeddability of a (finite) 3nilpotent semigroup in a finite completely 0simple semigroup is decidabl ..."
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The undecidable problems of the title are concerned with the question: is a given finite semigroup embeddable in a given type of completely 0simple semigroups? It is shown, for example, that the embeddability of a (finite) 3nilpotent semigroup in a finite completely 0simple semigroup is decidable yet such embeddability is undecidable for a (finite) 4nilpotent semigroup. As well the membership of the pseudovariety generated by finite completely 0simple semigroups (or alternatively by finite Brandt semigroups) over groups from a pseudovariety of groups with decidable membership is shown to be decidable. 1