The lattice of equational theories. Part IV: Equational theories of finite algebras
Jaroslav Ježek · Czechoslovak Mathematical Journal · 1986
This paper is a continuation of [1], [2] and [3].The lattice J5f j of equational theories of type A is antiisomorphic to the lattice of varieties of zl-algebras.The variety, corresponding to an equational theory T, is denoted by Mod(T); its elements are called models of T. If X is any class of J-algebras, then Eq(X) denotes the equational theory corresponding to the variety HSP(K) (the variety generated by K).For any algebra Ä put Eq(y4) == Eq({^}); this equational theory is called the equational theory of Л; it is just the set of equations satisfied in the algebra Ä.In this paper we shall be interested in the equational theories of finite algebras.Our aim is to prove that for any type A, the set of the equational theories of finite J-algebras is definable in the lattice J^^ and that in the case of a finite type A, the equational theory of any finite J-algebra is definable up to automorphisms in J^^.This will answer a problem formulated by George McNulty.For this purpose, we shall have to find a suitable encoding of finite algebras in ^^.The formulas ij/^o ^^^ ^45? the two most important formulas discovered in [3], enable us to carry most of the work over from ^^ to the lattice J^j of full sets of J-terms.And so instead of in J^j we shall encode the algebras in J^j.We shall not confine ourselves to finite algebras: in the case of a strictly large type A all algebras of cardinality ^Max(Ko, Card(zl)) will be encoded, while in the case of a large but not strictly large type the same will be done for the algebras of cardinahty ^ Max(Ko, Card(zl \^o)) only.For the terminology and notation see [1], [2] and [3].Algebras are often identified with their underlying sets.If Л is a ^-algebra and F A is a symbol of an arity n, then the corresponding л-агу operation in Ä will be denoted by F^.Most of the lemmas are without proof; they are either evident or follow easily from the preceding ones.I would hke to correct one wrong place in Section 5 of [2]: the definition of the