The congruence lattice of a regular semigroup

Semigroups and their lattice of congruences II

Joins and meets of congruences on a regular semigroup

On Congruence Lattices of Regular Semigroups

The class of 0-modular bands of groups is defined by means of a type of modularity condition on the lattice of congruences on a band of groups. The main result characterizes d-modularity as a condition on the multiplication in the band of groups. This result is then applied to the classes of normal bands of groups and orthodox bands of groups. Introduction. The class of 0-modular bands of groups is defined in 11u, where it is shown that 0-modularity is necessary and sufficient for a certain function to embed the lattice of congruences A(S) in a product lattice. In this paper, we investigate some properties of 0-modular bands of groups, and find several conditions equivalent to 0-modularity. The main result gives a condition on the multiplication in a band of groups S which is necessary and sufficient for S to be 0-modular. This result is then applied to the classes of normal bands of groups and orthodox bands of groups, to obtain a very simple characterization of 0-modularity in these classes. 1. Preliminaries. We use the notation and terminology of Clifford and Preston [21, with the following exceptions: x1: the inverse of x in H5, in a band of groups. B(S): the lattice of band congruences on S. M(S): the lattice of idempotent-separating congruences on S. t (x, y)}*: the congruence generated by the relation whose only element is the pair (x, y) (see [2, Theorem 1.81). By a band of groups, we mean a union of groups on which Green's J{-relation is a congruence. Following [12i, a band B will be called normal if axya = ayxa for all a, x, y E B. Theorem 8.2.9 of [61 lists several necessary and sufficient conditions for a band to be normal. A band of groups S is called a normal band of groups if the band S/J{ is normal. It is known that a band of groups S is a normal band of groups if and only if S is a strong semilattice (mapping semigroup) of completely simple semigroups, namely the 93-classes of S. This result Presented to the Society, November 19, 1971 under the title The lattice of congruences on a band of groups; received by the editors April 21, 1972. AMS (MOS) subject classifications (1970). Primary 20M10, 20M15.

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Theta-Modular Bands of Groups

It is implicit in a result of Kapp and Schneider [3] that, if S isa completely simple semigroup, then the lattice Λ( S ) of congruences on S can be embedded in the product of certain sublattices. In this paper we consider the problem of embedding Λ( S ) in a product of sublattices, when S is an arbitrary band of groups. The principal tool is the θ-relation of Reilly and Scheiblich [7]. The class of θ-modular bands of groups is definedby means of a type of modularity condition on Λ( S ). It is shown that the θ-modular bands of groups are precisely those for which a certain function is an embedding of Λ( S ) into a product of sublattices. The problem of embedding the inverse semigroup congruences into a certain product lattice is also considered.

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Carl Spitznagel

Papers

The lattice of congruences on a band of groups