EXTERNALLY COMPATIBLE IDENTITIES IN PSEUDOCOMPLEMENTED DISTRIBUTIVE LATTICES
Katarzyna Hatkowska · Demonstratio Mathematica · 1987
EXTERNALLY COMPATIBLE IDENTITIES IN PSEUDOCOMPLEMENTED DISTRIBUTIVE LATTICESWe deal with generalisations of pseudooomplemented distributive lettioes in connection with externally compatible identities introduced by W. Chromik in [1].Let K be a variety of type ri T --N U {o}, where T is a nonempty set and £ denotes the set of all positive integers.By identities of type r we mean expressions of the form p = q, where p,q are n-ary polynomial symbols of type r for some neNu{o} (see [2]).An Identity p • q is called externally compatible if it is of the form x • x or of the form f^(p 1 ,...j) --f t^-)» ,, *' (J r(t)' for S0B0 P ol 7 nomial symbols p., ,...,p r ^t), (t) 80me ^^^sntal operation symbol f t (see [1]), If K is a variety of algebras of type r, then S(K) (Ex(K)) denotes the set of all identities (all externally oompatible identities) satisfied in K.If S is a set of identities of type r, then V(S) denotes the variety defined by 3.In [1] representation theorems for algebras of V>(5x(E)) are given for some varieties K, namely, when K is an idempotent variety or K is a variety with unary operation symbol f suoh that f(f(x)) «= xbelongs to B(K) and any operation symbol g different from f is idempotent.Sxamples of such olasses are the class of all distributive lattices and the olass of all Boolean algebras.Moreover,the finite equational base for V(Ex(K)J in these both oases is given.