Complete endomorphisms of the lattice of pseudovarieties of finite semigroups

Abstract

The main result established that the mapping V → V ∩ W (V ∈ ℒ(F)) is a complete endomorphism of the lattice ℒ(F) of pseudovarieties of finite semigroups for certain particular pseudovarieties W, including the pseudovariety of bands.

The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

“…This result was proved very recently, and independently from our work, by Reilly and Zhang [30]. We give another proof of this result in this paper, but we prove in fact a much more precise result: for each pseudovariety of bands Z (given, say, by a defining band identity), we give an explicit formula for the subpseudovariety Z J of DA such that, for each subpseudovariety X of DA, we have (Note that Reilly and Zhang's result is wider in scope, since they consider the mapping V V S B on the lattice of all pseudovarieties of finite semigroups; but it is less precise, since they only prove the existence of the pseudovarieties Z J .)…”

“…Until recently however, the interaction between the techniques and results developed in the study of these two lattices was only minimal. This situation has changed in the last couple of years, for instance in Reilly [29], Auinger, Hall, Reilly and Zhang [5], and Reilly and Zhang [30].…”

“…Until recently however, the interaction between the techniques and results developed in the study of these two lattices was only minimal. This situation has changed in the last couple of years, for instance in Reilly [29], Auinger, Hall, Reilly and Zhang [5], and Reilly and Zhang [30].This paper is a contribution to the effort of using results on varieties of completely regular semigroups to study pseudovarieties of finite semigroups. Here we choose a restricted framework, namely that provided by the lattice of band Presented by B. M Schein.…”

Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands

“…In [24], the authors showed that the mapping χ : pseudovarieties of finite bands. Now the lattice L (B) of varieties of bands has a well known structure, each variety of bands has a finite basis (Birjukov [4], Fennemore [7], Gerhard [8]) and each variety of bands is generated by its finite members so that the lattices L(B) and L(B) are isomorphic.…”

Profinite Identities for Finite Semigroups Whose Subgroups Belong to a Given Pseudovariety

“…Thus, Corollary 4.7 is to be treated rather as a confirmation of the language of pro-identities being adequate from the computational complexity point of view than as a proper way towards an efficient solution of the membership problem for H. It is worth noting, however, that there is a natural class of pseudovarieties for which polynomial algorithms obtained via Theorem 4.5 are so far the only known tool for efficient membership testing. Namely, as it follows from a result by Reilly and Zhang [25], for every pseudovariety V of finite bands, there exists a greatest pseudovariety V f of finite semigroups such that V f ∩B = V , where B stands for the pseudovariety of all finite bands. One may observe a certain analogy between the pseudovarieties V f and H because the latter can be defined in a similar fashion as a greatest pseudovariety of finite semigroups such that H ∩ G = H, where G is the pseudovariety of all finite groups.…”