Semigroup Forum Vol. 44 (1992) 1-8 9 1992 Springer-Verlag New York Inc. RESEARCH ARTICLE Normal Skew Lattices Jonathan Leech Communicated by B. Schein Recall that a band S is normal if each principal monoid eSe is a semilattice. Equivalently, a normal band is a band satisfying the identity, wxyz = wyxz. Recall also that a skew lattice is an algebra (S,V, A) such that V and A are associative, idempotent binary operations on the set S which are connected by the absorption laws, x A (x V y) = x = (y V x) A x, and their duals. By a normal skew lattice we mean a skew lattice (S,V,A) such that each principal subalgebra x A S A x is a sublattice of S. Skew lattices naturally arise as multiplicative bands of idempotents in rings. In particular, every maximal normal band of idempotents in a ring forms a normal skew lattice which is the full set of idempoten~ in the subring it generates; and conversely, when the idempotents of a ring are closed under multiplication, they form a normal skew lattice. (See [6] 2.2.) Upon examination of the semigroup ring of a normal band, one thus obtains that every normal band can be embedded in a normal skew lattice. Hence there is a sense in which normal skew lattices form completions of normal bands; as such, the theory of normal skew lattices may be seen to extend the theory of normal bands initiated in [9], [12], and [13]. This paper is divided into three sections, the first of which is a prelim- inary section giving the skew lattice analogues of some basic results of Yamada and Kimura. The remaining sections focus attention on the principal theme of this paper, the connection between normality and distributivity. The main result of the middle section, Theorem 2.8, gives a canonical factorization of a normal symmetric skew lattice into the fibered product of a lattice with a distributive skew lattice; this effectively reduces the study of normal symmetric skew lat- tices to the distributive case. The distributive case is considered in the final section which contains analogues of several fundamental results about distribu- tive lattices. Finally, the needed background on bands, distributive lattices, and universal algebras may be found in [2], [3], and [8]. 1. Basic Theory 1.1. Recall that a rectangular skew lattice is a skew lattice S for which (S, A) is a rectangular band and (S, V) is its dual where x V y -- y A z. The first important theorem on skew lattices is the Clifford-McLean Theorem: the maximal rectangular subalgebras of a skew lattice form a congruence partition for which the induced quotient algebra is the maximal lattice image. The congruence classes are called equivalence classes and two members, x and y, of the same class are said to be equivalent, denoted x - y. The natural partial ordering of a skew lattice is defined by x _ y iff x A y -- y = y A x, or dually, x V y -- x -- y V x. Our goal is to show how a normal skew lattice is constructed from its equivalence classes. For normal bands there is the construction of