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

Abstract

We use classical results on the lattice L(B) of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice Ps(DA) of subpseudovarieties of DA, -where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. We bring forward a lattice congruence on Ps(DA), whose quotient is isomorphic to L(B), and whose classes are intervals with effectively computable least and greatest members. Also we characterize the pro-identities satisfied by the members of an important family of subpseudovarieties of DA. Finally, letting V k be the pseudovariety generated by the k-generated elements of DA (k ] 1), we use all our results to compute the position of the congruence class of V k in L(B). IntroductionThe lattice of pseudovarieties of finite semigroups has been the object of much attention over the past few decades, with motivations drawn not only from universal algebra, but also from theoretical computer science. The link between theoretical computer science and semigroup theory goes back to Schü tzenberger's work in the late 1950s, and the role played there by pseudovarieties of finite semigroups was detailed by Eilenberg in the 1970s [7]. For more recent developments, the reader is referred to the books by Pin [21] and Almeida [1].Independently of this research, the lattice L(CR) of varieties of completely regular semigroups also received considerable attention, through the work of Polák [25,26,27], Reilly [28], Pastijn and Trotter [18], and others. 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

“…Since RB is given by a finite basis of identities, it follows that we can find by the above discussion a finite basis of pseudoidentities for every pseudovariety of the form V drb . Independently, Trotter and Weil [29] have considered this case using quite different techniques.…”

“…For a comprehensive bibliography, the reader is referred to Petrich and Reilly [19]. A series of recent papers, Reilly and Zhang [25], Auinger, Hall, Reilly and Zhang [3], Auinger [2], Hall and Weil [11], Hall and Zhang [12], Pastijn and Trotter [18] and Trotter and Weil [29], has explored the extension of these ideas, first to the lattice of existence varieties of regular semigroups and then to the lattice of pseudovarieties of finite semigroups. Some of these complete congruences on L(CR) are induced by mappings of the form V −→ V ∩ X for some special variety X , such as the variety of groups, completely simple semigroups or bands (see Jones [15], Trotter [28] or Reilly [23]).…”

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

“…As the name implies, this hierarchy was first studied by Trotter and Weil [28]. We will define it using Mal'cev products.…”

Representations of the Free Profinite Object Over Da

“…We define the central basic factorization of w (see Almeida [3] or Trotter and Weil [15]) as a factorization of one of the following forms:…”