J. A. Gerhard

The lattice of equational classes of idempotent semigroups

The concept of an equational class was introduced by Birkhoff [2] in 1935, and has been discussed by several authors (see for example Tarski [22]). Recently there has been much interest in the subject, and especially in the lattice of equational classes of lattices [l], [8], [I 11, [12], [17]. However, until now the only nontrivial lattice of equational classes to be completely described was the lattice of equational classes of algebras with one unary operation (Jacobs and Schwabauer [IO]). The problem of describing the lattice of equational classes of algebras with a single binary operation, i.e., groupoids. is much more difficult. Kalicki [13] h as shown that there are uncountably many atoms in the lattice of equational classes of groupoids. The lattice of equational classes of semigroups (associative groupoids) is uncountable (Evans [5]) and has been investigated by Kalicki and Scott [14], who listed its countably many atoms. There are only two equational classes of commutative idempotent semigroups, but the removal of either commutativity or idempotence gives rise to a non-trivial lattice. Partial results have been obtained for the lattice of equational classes of commutative semigroups [19], [20]. In the case of idempotent semigroups it is relatively easy to show that the lattice of equational classes has three atoms, and in fact the sublattice generated by the atoms has been shown by Tamura [21] to be the eight-element Boolean lattice. Kimura [ 151 has described all equations on idempotent semigroups which have at most three variables. In this paper a complete description of the lattice of equational classes of idempotent semigroups is given. We consider the elements of the lattice to be the equational classes themselves and ignore the foundation problem which this entails. In any case this problem can be easily circumvented, for example by taking as elements of the lattice the fully invariant congruences of the free idempotent semigroup on countably many generators.

Subdirectly irreducible idempotent semigroups

Subdirectly irreducible idempotent semigroups were characterized in [3], and in that paper, their connection with the various equational classes of idempotent semigroups was discussed. All these results are in terms of identities, so that examples of subdirectly irreducibles in the equational classes are explicitly known only for small classes. It is easy to show from general considerations (see the last section of the present paper) that every proper equational subclass of the class of idempotent semigroups is generated (as an equational class) by one or two subdirectly irreducibles. In this paper we give an example of a subdirectly irreducible for each join irreducible equational class of idempotent semigroups, which generates the class. This list, together with known results, gives explicit examples of one or two finite subdirectly irreducibles which generate the various equational classes.

Free completely regular semigroups I. Representation

The free completely regular semigroup on a set X has been studied by Clifford [2]. He has discussed properties of these semigroups for general X and given their structure in case 1x1~ 2. (For 1x1 = 1, the free completely regular semigroup is just the free group.) In this series of papers (parts I and II), we give the structure of the’ free completely regular semigroup on X and in fact solve the word problem for such semigroups. Part I gives a representation for the free semigroup on X using the description of an arbitrary completely regular semigroup given by Petrich in 14, Theorem 31. This representation is of some independent interest and lays the groundwork for the solution of the word problem given in past II.

The word problem for orthogroups

A semigroup which is a union of groups is said to be completely regular. If in addition the idempotents form a subsemigroup, the semigroup is said to be orthodox and is called an orthogroup. A completely regular semigroup 5 is provided in a natural way with a unary operation of inverse by letting a~ for a Ç S be the group inverse of a in the maximal subgroup of S to which a belongs. This unary operation satisfies the identities

Unification in free distributive lattices

Abstract For a pair of words in any free distributive lattice, a procedure is given which decides whether they can be unified. Bases for all unifiers are found in the case where the two words to be unified are ∧-words (or by duality ∨-words). The cases where the total number of generators is at most three are discussed in detail. This includes all the possible combinations of constants and variables. Bases for the unifiers are given in all these cases.