Matthew Gould

Boolean extensions and normal subdirect powers of finite universal algebras

(i) ~B is subdirectly representable in {91o . . . . ,91,_2}c_K if and only if satisfies the identities of 91o x ... x 91n2. (ii) I f I f3 ] > 1 and ~ is subdirectly representable in K, then ~ has a unique set of factors in K. We shall generalize the concept of normal subdirect power in a way that will enable us to prove Theorems 1 * and 2* for arbi t rary (rather than binary or f-) algebras; Theorem 3 will be proved without recourse to the theory of f algebras.

Posets whose monoids of order-preserving maps are regular

The purpose of this paper is to determine all posetsP such that the monoid of all order-preserving maps ofP intoP is regular in the semigroup-theoretic sense.

Globally determined lattices and semilattices

whenever f is an n-ary operation in F and A1,.. . , A n are complexes of A. A class Yg of algebras is said to be globally determined if any two members of having isomorphic globals must themselves be isomorphic. Tamura and Shafer [9] noted that the class of all groups is globally determined, and from the easy proof of this fact follows the corresponding result for rings. Certain classes of semigroups, including the class of all finite semilattices, were shown by Gould and Iskra [2] to be globally determined. Such results were established for various classes of semigroups by Tamura [8], Tamura and Sharer [10], Va~enin [11], and Mogiljanskaja [3] [4] [5], who also exhibited [6] [7] pairs of non-isomorphic infinite semigroups having isomorphic globals. The fact that finite mono-unary algebras are globally determined was established by Drgtpal [1], along with a counterexample for the infinite case. We shall show that the class of all semilattices with identity is globally determined (Theorem 1.3) and utilize this result to prove that the class of all lattices is globally determined (Theorem 2.2).