New Classes of Ideals in Subtraction Algebras

In [4] B. M. Schein considered systems of the form ( ; o; n),where is a set of functions closed under the composition "o" of functions (and hence ( ; o) is a function semigroup) and the set theoretic subtraction "n" (and hence is a subtraction algebra in the sense of [1]). He proved that every subtraction semigroup is isomorphic to a di erence semigroup of invertible functions. B. Zelinka [5] discussed a problem proposed by B. M. 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. A particular attention was paid to the ideals of a subtraction algebra (cf. [2, 3] and papers cited there). Prime ideals play a central role in the theory of subtraction algebras. Of course, a prime ideal P of a subtraction algebra X is a proper ideal of X with the property that for a; b 2 X, a ^ b 2 P implies a 2 P or b 2 P . In this paper we give some generalizations of prime ideals: weakly prime ideals and 2-absorbing ideals. We introduce also new classes of ideals of X: Primal ideals and weakly primal ideals. Then we give some basic results about these classes of ideals.