Introduction to Relation Algebras pp 313-383 | Cite as

# Ideals and quotients

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## Abstract

Another way of constructing a new relation algebra from a given one A is to “glue” some of the elements of \( \mathfrak{U} \) together to form an algebra that is structurally similar to, but simpler than \( \mathfrak{U} \). Congruence relations provide a natural way of carrying out such a gluing. The prototype for this construction is the ring of integers modulo *n* (for some positive integer *n*), which is constructed from the ring of integers by forming its quotient with respect to the relation of congruence modulo *n*. Each congruence relation on a relation algebra determines, and is determined by, the set of elements that are congruent to zero, so the whole construction can be simplified by replacing the congruence relation with the congruence class of zero. This leads to the study of ideals. The most natural place to begin the discussion, however, is with the basic notion of a congruence relation.

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