The join of the pseudovarieties of idempotent semigroups and locally trivial semigroups

This article solves a problem proposed by Almeida: the computation of the join of two well-known pseudovarieties of semigroups, namely the pseudovariety of bands and the pseudovariety of locally trivial semigroups. We use a method developed by Almeida, based on the theory of implicit operations. This work was partly supported by PRC Mathématiques et Informatique and by ESPRITBRA WG 6317 ASMICS-2     

The joint of the pseudovarieties of idempotent semigroups and locally trivial semigroups

Communicated by J.-E. Pin     

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

It has been shown by the authors that the mapping,where B is the pseudovariety of finite bands, is a complete retraction of the lattice L(F) of pseudovarieties of finite semigroups onto the lattice of pseudovarieties of bands. It follows that the classes of the induced congruence on L(F), or on the lattice of subpseudovarieties L(W) for any subpseudovariety W of F, are intervals. In this paper we solve the membership problem for the upper limit of the classes of restricted to L(W) for various , including F itself, and provide bases of pseudoidentities for certain cases. Received April 7, 1999; accepted in final form April 30, 2000.     

Minimal non-permutative pseudovarieties of semigroups III

Various types of permutativity conditions are important in the study of (pseudo)varieties of semigroups. In previous papers of this series, minimal pseudovarieties failing certain permutativity conditions were determined. This theme is pursued here in relation to a family of conditions which appear in the investigation of power pseudovarieties of semigroups, namely, for eachk0, we consider the conditionC _k on a semigroup that it satisfy the identityx ₁ ...x _k y ₁ y ₂ z ₁ ...z _n=x ₁ ...x _k y ₂ y ₁ z ₁ ...z _n for somen. For all minimal pseudovarieties failing C_k, a generator and finite basis of identities are specified.This work was supported, in part, by NSERC Grant A4044.Presented by Boris Schein.     

On semidirectly closed pseudovarieties of aperiodic semigroups

The aim of this work is to study the unknown intervals of the lattice of aperiodic pseudovarieties which are semidirectly closed and answer questions proposed by Almeida in his book “Finite Semigroups and Universal Algebra”. The main results state that the intervals [V^*(B₂),ER∩LR] and [V^*(B₂¹),ER∩A] are not trivial, and that both contain a chain isomorphic to the chain of real numbers. These results are a consequence of the study of the semidirectly closed pseudovariety generated by the aperiodic Brandt semigroup B₂.     

Demidirectly closed pseudovarieties of locally trivial semigroups

Communicated by N.R. Reilly     

The lattice of idempotent binary relations

Order-theoretic properties of the complete latticeE(A) of indempotent binary relations ρ=ρ² on the given setA are investigated. The elements ρ ofE(A) are classified according to theirfixed field I(ρ)={a∈A|(a, a)∈ρ} as being either offinite type, dense, or ofmixed type. When |A|>1E(A) is a non-atomic, non-coatomic lattice in which each element is a meet of meet-irreducible elements. The elements ofE(A) which are joins of join-irreducible elements form a compactly generated complete latticeF(A) which is a join-sublattice ofE(A) consisting of all elements having finite type. The setsD(A), M(A) of elements ρ ofE(A) which are dense (i.e., satisfy ρ≠ϕ andI(ρ)=ϕ) or of mixed type (i.e., are neither dense nor of finite type) resp. are non-empty only when |A| is infinite.D(A) is a partial meet-subsemilattice ofE(A) admitting no minimal elements. The group of order automorphisms of the latticeE(A) is isomorphic toS _A ×Z ₂ and each order automorphism ofE(A) preserves inverses. Presented by B. M. Schein.     

Global determinism of idempotent semigroups

The power semigroup, or global, of a semigroup S is the set 𝒫(S) of all nonempty subsets of S equipped with the naturally defined multiplication. A class 𝒦 of semigroups is globally determined if any two members of 𝒦 with isomorphic globals are themselves isomorphic. The principal goal of this paper is to prove that the class of all idempotent semigroups is globally determined.     

有限链上矩阵半群中幂等元的刻划

利用幂等元的定义以及矩阵的乘法研究了有限链K上的n×n阶矩阵的乘法半群Mn(K)中的幂等元,并给出若干刻划,主要是Simon结果的推广.从而,就可将Fuzzy矩阵中某些问题的研究转换为有限链K上矩阵中相应问题的研究,使其简单     

The lattice of idempotent distributive semiring varieties

A solution is given for the word problem for free idempotent distributive semirings. Using this solution the latticeL (ID) of subvarieties of the variety ID of idempotent distributive semirings is determined. It turns out thatL (ID) is isomorphic to the direct product of a four-element lattice and a lattice which is itself a subdirect product of four copies of the latticeL(B) of all band varieties. ThereforeL(ID) is countably infinite and distributive. Every subvariety of ID is finitely based. Project supported by the National Natural Science Foundation of China (Grant No. 19761004) and the Provincial Applied Fundamental Research Foundation of Yunnan (96a001z).     

Unification in varieties of idempotent semigroups

We have classified all varieties of idempotent semigroups with respect to the unification types of their defining sets of identities. With the exception of eight finitary unifying theories these are all of unification type zero. This yields countably many examples of theories of that type which are more “natural” than the first example constructed by Fages and Huet [9, 10].