Generalized seminear-fields

Abstract

A triple (S, +, ·) is said to be a seminear-ring iff (1) (S, +) and (S, ·) are semigroups and (2) (x + y) z = xz+yz for all x, y, z {u1D4D4} S. A seminear-ring (D, +, ·) is said to be a ratio seminear-ring iff (D) is a group. A seminear-ring (K, +, ·) is said to be a generalized seminear-field iff there exist an element a in K such that (K ̀ {a}, ·) is a group, and such an element a is called a special element of K. In this study we found that a special element of a generalized seminear-field has 6 types as the following theorem shows: Theorem 1 Let k be a generalized seminear-field with a as a special element. Then exactly one of the following statements hold: (1) ax = xa = a for all x {u1D4D4} K. (2) ax = xa = x for all x {u1D4D4} K. (3) ax = a and xa-x for all x {u1D4D4} K. (4) ax = x and xa = a for all x {u1D4D4} K. (5) a2 ≠ e and ae = ea = a. (6) a2 ≠ e and ae = ea ≠ a where e is the identity of (K ̀ {a}, ·). In case (6) we found that there exists a unique element d in K ̀ {a} such that ax = dx and xa = xd for all X {u1D4D4} K. ___ (*) A special element a satisfying (1), (2), (3), (4), (5) or (6) is called a category I, II, III, IV, V or VI special element of K, respectively. Theorem 2 Let k be a generalized seminear-field with a as a special element. If a is a category II or VI special element of K then (K ̀ {a}, +, ·) is a ratio seminear-ring. Given a ratio seminear-ring D we classifield all generalized seminear-fields K with a category II special element a such that D = K ̀ {a}. Given a ratio seminear-ring D and d {u1D4D4} D we classified all generalized seminear-fields K with a category VI special element a such that D = K ̀ {a} and d satifies (*). We also classified all generalized seminear-fieldes containing a category V special element.