SUBDIRECT SUM OF SUBTRACTION ALGEBRAS

Abstract

Abstract. Subdirect sum of algebras are introduced,and related properties are investigated. 1. IntroductionB. M. Schein [9] considered systems of the form ('; –;n ), where ' isa set of functions closed under the composition – of functions (andhence ('; – ) is a function semigroup) and the set theoretic subtraction n (and hence ('; n ) is a algebra in the sense of [1]). Heproved that every semigroup is isomorphic to a difierencesemigroup of invertible functions. B. Zelinka [10] discussed a problemproposed by B. M. Schein concerning the structure of multiplication in asubtraction semigroup. He solved the problem for algebrasof a special type, called the atomic algebras. Y. B. Jun et al.[4] introduced the notion of ideals in algebras and discussedcharacterization of ideals. In [3], Y. B. Jun and H. S. Kim establishedthe ideal generated by a set, and discussed related results. Y. B. Jun andK. H. Kim [5] introduced the notion of prime and irreducible ideals of asubtraction algebra, and gave a characterization of a prime ideal. Theyalso provided a condition for an ideal to be a prime/irreducible ideal. Y.H. Kim and E. H. Roh [6] introduced the notion of neutral subtractionalgebras and neutral ideals, and investigated several properties. Also, E.H. Roh [7], [8] developed the theories of prime ideals and completed insubtraction algebras, and investigated several properties. In this paper,we introduce the notion of the subdirect sum in algebras.