Lattices of completely regular semigroup varieties

Francis Pastijn and Peter George Trotter

Vol. 119 (1985), No. 1, 191–214

DOI: 10.2140/pjm.1985.119.191

Abstract

Let ρ be a fully invariant congruence on the free completely regular semigroup F_X^CR of countably infinite rank. Let ρ_min and ρ^min be the least congruences on F_X^CR with respectively the same trace and the same kernel as ρ. Let ρ_max and ρ^max be the greatest congruences on F_X^CR with respectively the same trace and the same kernel as ρ. These congruences are shown to be fully invariant. We construct a network of varieties corresponding to the congruences ⋯,ρ^max,ρ_max,ρ,ρ^min,ρ_min,(ρ_min)^min,(ρ^min)_min,⋯ and their intersections. Intervals between successive joins of the network, in the lattice of subvarieties of completely regular semigroups, are characterised as direct products of particular subintervals. By comparing the network with the chain of varieties that are each generated by a free completely regular semigroup of finite rank we get information on the network and the chain.

Mathematical Subject Classification 2000

Primary: 20M07

Secondary: 08B15

Milestones

Received: 29 December 1983

Published: 1 September 1985