Construction of the Smallest Common Coarser of Two and Three Set Partitions

Abstract

This paper is inspired by a text of the book [7] ("Úvod do algebry" in Czech, "Introduction to Algebra" in English) of the authors Ladislav Kosmák and Radovan Potůček. They followed the great work of Professor Otakar Borůvka in the field of the partition theory, groupoids and groups and gave them in the context to contemporary modern algebra. Academian Borůvka have deduced and proved many results concerning the partition theory in his publications.His first works [1] and [2] were published during World War II and his monographs [3] and [4] were released in the post-war years.In this paper we deal with a construction of the smallest common coarser of two set partitions associated with equivalence relations, we give a special relation used in the construction and an illustration of blocks of this coarser.Remark 1. Algebra of binary relations was worked up by Jacques Riguet in 1948 (see [8], the translation into Russian has its origin from 1963). Professor Borůvka was rightfully proud himself on his theory of partitions and an appliance of the algebra of relations has never used. From twenty-seven chapters of the book [4] is six devoted to the general theory of partitions and in the chapter 14 and 21 -24 are paragraphs dealing with partitions in the theory of grupoids and groups.

Decomposition with Respect to Outputs for Boolean Control Networks

“…(37) Let us briefly recall some basic and well-known terms, notions and facts, without proofs, concerning partitions and equivalence relations. For details, please refer to Borůvka (1974) and Potůček (2014).…”