Varieties of implicative semi-lattices. II
William C. Nemitz, Thomas P. Whaley · Pacific Journal of Mathematics · 1973
This paper is concerned with a process of coordinatization of the lattice of varieties of implicative semilattices.Equational descriptions of some elements in each coordinate class, and a complete equational description of one coordinate class are given.1* Introduction* This paper is a continuation of [8] Familiarity with [8] and [6] is assumed.After stating some of the consequences of the local ίiniteness of the variety of implicative semilattices, we describe a system for partitioning the lattice of varieties of implicative semi-lattices into coordinate intervals, and give some results that can be obtained from a study of this coordinatization.Finally, we give equational descriptions for the largest and smallest varieties in each coordinate class, the covers of the smallest variety in each coordinate class and a complete equational description of the coordinate class 4,2.Recall that an implicative semi-lattice is subdirectly irreducible if and only if it has a single dual atom.In accordance with the usage of [8], this dual atom will be denoted by u.If in a subdirectly irreducible implicative semi-lattice, the dual atom is deleted, the remaining structure is both a subalgebra and a homomorphic image of the original.Thus every subdirectly irreducible implicative semi-lattice may be thought of as obtained by appending a single dual atom to some already given implicative semi-lattice.If L is an implicative semilattice, the subdirectly irreducible implicative semilattice obtained in this manner will be denoted by L.2* Local finiteness* The following theorem was proven first by A. Diego [2] in a slightly different context.McKay [4] extended the result to implicative semi-lattices.We present a much simpler proof here.THEOREM 2.1.The variety of implicative semi-lattices is locally finite.304 W. NEMITZ AND T. WHALEYof which is finite.Therefore the same statement applies to the L i9 and hence F n+1 is finite. COROLLARY 2.2. Every variety of implicative semi-lattices is generated by its finite sub-directly irreducible members.COROLLARY 2 3. // / is a homomorphism of an implicative semilattice L onto a finite implicative semi-lattice M, then there exists U ^8L such that f\U is an isomorphism.COROLLARY 2.4.The lattice of all varieties of implicative semilattices is itself implicative.COROLLARY 2.5.If L is a finite subdirectly irreducible implicative semi-lattice, then the class of all those implicative semilattices which do not contain a sub-implicative semi-lattice isomorphic to L is a variety.COROLLARY 3.2.V n , m contains only a finite number of distinct finite subdirectly irreducible algebras.