Partitions and congruences in algebras. IV. Associable systems
Tran Duc · Czech digital mathematics library · 1974
We recall briefly some definitions and results with which it is possible to get acquainted in the introductory paragraphs of the paper [11].The notion of partition on a set was studied in many papers, e.g.[1,3,4, 6,7,8,9,10,13,14].A partition in a set G is a partition on a subset of the set G. The elements of the partition are called blocks and the union u A of all blocks of the partition A is called the domain of the partition A. The set P(G) of all partitions in the set G is in a one-to-one correspondence with the set of all symmetric and transitive binary relations (ST-relations) in G.The papers [2,5, 11,12] deal with the partitions "in", to a smaller extent also [3,4].Under the congruence in a universal algebra (G, Q) we understand the stable ST-relation in the algebra (G, Q).By the symbol JT(G) we denote the lattice of all congruences in (G, Q) 9 symbols v^, Vr mean the supremum in the lattice Jf(G).As a rule, however, we write simply v, V instead of v P , \f P for the supremum in the lattice P(G).4.0 In the paper [8] the concept of associable system of partitions on a set was introduced and in [6] it was generalized for the partitions in a set.In [14] the term of "absolutely permutable system of relations of equivalence" was used for the same concept; see also [7].This concept represents a generalization of the permutability of partitions in a set for a system {A l9 A 2 } of two partitions in a set G is associable if and only if A l9 A 2 commute [6] Lemma 1.2, see also 4.3.In this paper we consider a system of congruences in an algebra G associable if the corresponding system of partitions in G is associable.Many theorems in the sequel use and generalize results of the paper [6] on partitions and apply them to the congruences in algebras, especially in O-groups.Main general results for the partitions are included in Theorems 4.17, 4.19 and 4.22. Definition.A system {A ( : i e F} of partitions in a set G is called associable if it satisfies: For any system $t = {x t : ieT} of elements of the set G fulfilling x a ( V -4,) y? (oc, peF) there holds one of the following conditions: (4.1,1) x e G exists such that x'A.x, i e F (4A,2) a e F and A\ e A a exist such that 21 g A* and if A\ n u A p 7-= 0 for some peT 9 then A\eA$.([6], Definition 1.2)