The largest groupoid of varieties of semigroups with zero

Abstract

It is well known [i] that with respect to the operation of multiplication, the set of all varieties of semigroups forms a partial groupoid. The structure of this is very complicated and still not known. In studying this groupoid the most natural problems are the following: i) to describe all pairs of varieties whose product is a variety; 2) to find in this partial groupoid some important groupoids (in particular, semigroups) and to clarify their structure; 3) to distinguish maximal groupoids and semigroups. Some of these investigations have already been carried out. Thus, in [2] idempotents of this groupoid were described and a countable semigroup with nullary multiplication was distinguished, which consists of varieties of indempotent semigroups; in [3, 4] some conditions were given under which the product of two varieties of semigroups is a variety; in [5] a groupoid with the power of the continuum was mentioned, which consists of so-called 0-reduced varieties, that is, varieties of semigroups with zero O, having an identity basis of the form w = 0, where w is a semigroup word; in [6] a number of important properties of this groupoid G were proved: it is cancellative and is the union of two nonintersecting subgroupoids H and L, where H is the largest subsemigroup, and L is an ideal of G. The structure of H has been completely clarified [7, 8]: it is the free product of a free cormmutative semigroup of countable rank and a free semigroup of the rank of the continuum with externally adjoined unity. The main result of the paper is the following theorem. THEOREM. The groupoid G is the maximal groupoid in the partial groupoid of all varieties of semigroups. In the partial groupoid of varieties of semigroups with signature zero, G is the largest groupoid. We mention that this theorem gives a complete answer to the third question for the partial groupoid of semigroups with signature zero. Before proceeding to the proof of the theorem, we give the necessary definitions and notation. In this paper we use the generally accepted terminology (see [9, i0]). Our notation is also generally standard or follows the notation in [6]. Let X be a countable alphabet, and F a free semigroup over X. For a word a of F we denote by C(a) the set of all letters of X that occur in writing 0~, and by 151 the length of the word ~L. A letter that occurs more than once in writing OJ will be called multiple, and the letters that appear in the first and last places in writing O~ will be called the beginning and end of ~L. For the graphic coincidences of the words C5 and b of F we shall write ~L=---~. The notation ~dL means that b is a subword of /~; ~ <0~ means that ~ is