DEMONSTRATIO MATHEMATICA Vol. XXVII No 3-4 1994 Κ. Denecke PRE-SOLID VARIETIES Dedicated to Professor Tadeusz Traczyk 1. Introduction An identity t « t' of terms of any type τ is called a hyperidentity for a universal algebra A = ( A ; (/• A ), e /) if t « t' holds identically for every choice of n-ary term operations to represent n-ary operation symbols occuring in t and t' ([8]). Although the concept of a hyperidentity is very strong there are countable infinitely many semigroup varieties for which every identity is a hyperidentity (solid varieties of semigroups) ([3]). Since any projection de- fined on A is a term operation of A, a hyperidentity must be satisfied at least for the projections. Therefore there are identities which cannot be hyperiden- tities. Substituting one of the binary projections for F in F(x, y) « F(y,x) we see that the commutative law fails to be a hyperidentity in any nontriv- ial variety with a binary operation symbol. This observation suggests the idea to weaken the concept of a hyperidentity. The simplest way for weake- ness could be to substitute only term operations different from projections. The set of all term functions of A which are different from projections can be regarded as the universe of an algebra whose fundamental operations describe the composition of functions, the so-called pre-iteratire algebra in the sense of I.A. Mal'cev ([6]). This motivates to denote these „weaker" hy- peridentities as pre-hyperidentities. An algebra or a variety for which every identity is a pre-hyperidentity is called pre-solid. After developing the the- ory of pre-hyperidentities and pre-solid varieties we will apply the results on semigroups and determine the greatest pre-solid variety of commutative semigroups. This paper has been presented at the Conference on Universal Algebra and its Appli- cations, organized by the Institute of Mathematics of Warsaw University of Technology held at Jachranka, Poland, 8-13 June 1993. Unauthenticated Download Date | 3/4/20 12:42 AM