Direct decompositions of lattices, I
Otomar Hájek · Czechoslovak Mathematical Journal · 1957
ТЫв article contains the foundations of the algebraical theory of direct and subdirect decompositions of lattices and rings.Except for theorem 14 and most of theorems 2 and 3, all non-trivial results are new.We shall, in general, use the notation of LT, with some exceptions.(LT means G. BIRKHOFF, Lattice Theory, 2nd.ed., New York, 1948.)In lattices, a is the set of x ^ a, a the set of x ^^ a, (a, 6) the set (interval) of a ^ x is implication; * in the text means end of proof.''Homomorphism" alwaj^s means lattice-homomorphism.For most of the elementary definitions use LT. Preliminary notionsDefinition. // S^ {aeA) are abstract algebras of the same type with oc-ary operations Z, then Pa^S^a ^^ ^^ß abstract algebra (direct product) consisting of all maps \Xa\a ' Ä. -> Ußa with Ж« € Ä^ , with a-ary operations I^ti^lla = l-^b^lla • (^^^ ^ finite A we shall, of course, use Si X S2 X ... X Sn, [Xi, ..., Xn], etc.; also, often, [Xa] instead of [Xala-)Definition. S = PaSa, 'read as "S is {decomposable into) the direct product of Sa's", means that there is an algebraic isomorphism^ between S cmd Pa^a^ *) n is the chain of n elements.