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

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 Boruvka 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.