N-IDEALS OF SUBTRACTION ALGEBRAS

Abstract

Using N-structures, the notion of an N-ideal in a subtrac- tion algebra is introduced. Characterizations of an N-ideal are discussed. Conditions for an N-structure to be an N-ideal are provided. The de- scription of a created N-ideal is established. A (crisp) set A in a universe X can be defined in the form of its characteris- tic function µA : X ! {0,1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A. So far most of the generalization of the crisp set have been conducted on the unit interval (0,1) and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive infor- mation that fit the crisp point {1} into the interval (0,1). Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al. (4) introduced a new function which is called negative-valued function, and constructed N-structures. They discussed N- subalgebras and N-ideals in BCK/BCI-algebras. Schein (6) considered systems of the form (';-,\), where ' is a set of functions closed under the composition - of functions (and hence (';-) is a function semigroup) and the set theo- retic subtraction (and hence (';\) is a subtraction algebra in the sense of (1)). He proved that every subtraction semigroup is isomorphic to a dierence semigroup of invertible functions. Zelinka (7) discussed a problem proposed by Schein concerning the structure of multiplication in a subtraction semigroup. He solved the problem for subtraction algebras of a special type, called the atomic subtraction algebras. Jun et al. (2, 3) introduced the notion of ideals in subtraction algebras and discussed characterization of ideals. Jun et al. (5) provided conditions for an ideal to be irreducible. They introduced the notion of an order system in a subtraction algebra, and investigated related proper- ties. They provided relations between ideals and order systems, and dealt with