CHAPTER THREE - UNIVERSAL ALGEBRAS GROUPS WITH MULTI-OPERATORS

Abstract

This chapter discusses how the concept of an isomorphism of universal algebras plays exactly the same part as it plays for groups, rings, or partially ordered set. The class of groupoids is a primitive class with respect to the binary operation and the empty set of identical relations. A skew field can be regarded as a universal algebra only in the case when the algebraic operations under consideration are assumed to be partial because this is the case with both left and right division in a skew field, and with taking the inverse of an element. There exists a very close connection between congruences and, hence, homomorphisms of groups and rings, on the one hand, and normal subgroups of groups and ideals of rings, on the other hand. This connection cannot be completely extended to the case of all universal algebras, in particular to the case of groupoids or semigroups.