On the fundamental theorem of lattice-primal algebras
A. Lenkehegyi · Institutional Repositories DataBase (IRDB) · 1985
It is known that in a variety generated by a lattice-primal algebra lJC (i.e., an ordered algebra with a lattice ordering, with respect to which the operations are monotonic and in which every finitary monotone function is a term-function) every element is isomorphic to a distributive extension lJC['.tl](i.e., extension by a bounded distributive lattice '.ti) of the generator algebra.Moreover, the variety as a category is equivalent to distributive lattices and antiequivalent to the category of all totally order-disconnected spaces, the so-called Priestley-spaces.This was essentially proved first by B. A. Davey, D. Duffus, R. W. Quackenbush and I. Rival.The results of our approach were proved independently two years later in 1980, and were announced at the Czechoslovakian Summer School in the same year.Our proofs seem to be more traditional and direct as they follow the lines of the corresponding earlier analogous results concerning varieties generated by primal algebras and Boolean extensions; but they are surely less elegant.Nevertheless, we still think that they reflect certain details better.These investigations were suggested by A. P. Huhn.We introduce normal subdirect powers of ordered algebras, they contain the diagonal and are closed under every monotone finitary function performed componentwise.After this we show, by J6nsson's lemma that in a lattice-primal generated variety every member is essentially (i.e., up to isomorphism) a (normal) subdirect power of the generator algebra.The crucial result is to prove that normal subdirect powers are all distributive extensions; we succeeded in giving a constructive proof for this.We also give two applications, first, we show that in a lattice variety generated by an orderfunctionally complete finite lattice 2, every member containing 2 as a "0, 1-sublattice" is isomorphic to a distributive extension of 2. Secondly, we give a proof for a result of G. Gierz telling that given a lattice 2 as before, gluing a distributive lattice '.ti into a prime interval [a, b] of 2, the free lattice over the variaty generated by 2, with the resulting partial lattice L~b as a free generating set, is siomorphic to 2['.tl], the extension of 2 by '.ti.A. L. Foster [1] introduced the concept of extending an algebra by a Boolean algebra: if &=(A; F) is an algebra, m=(B; /\, v, ', 0, 1) a Boolean algebra, then the extension of & by m is the algebra, whose underlying set consists of all maps e: A ➔ B, which are of finite range, a, beA, a#b implies e(a)Ae(b)=O, and V e( a)= 1.The operations are defined as follows: let f be an n-ary oper-aeA ation symbol from the type of~.then for any maps e 1 , ... , en satisfying the conditions just mentioned, J(e 1 , ... , en) is defined to be the map A ➔ B for which