Generalized semifields

Abstract

A triple (S, +, · ) is said to be semiring iff S is a set and + and · are binary operations on S such that (1) (S,+) is a commutative semigroup, (2) (S, ·) is commutative semigroup, (3) (x+y) ·z=x·z+y·z for all x, y, z, ɛ , S. A semiring (D,+, ·) is said to be a ratio semiring iff (D, ·) is a group. A semiring (K,+, ·) is said to be generalized semifield iff there exists an element a in K such that (K\{ a}, ·) is a group. In this thesis we still call gernerlized semifields “semifields” Theorem Let (K,+, ·) be a semifield and a an element in K such that (K\{ a}, ·) is a group. Then (ax = a for all x ɛ K) or (ax = x for all x ɛ K) or (ae ≠ a and a² ≠ a where e is the identity of (K\{ a}, ·)), From this theorem we have three types of semifields, they are: (1) ax = a for all x ɛ K (called type I semifields w, r ,t a), (2) ax = x for all x ɛ K (called type II semifields w, r, t, a), (3) a·e ≠ a และ a² ≠ a where e is the identity of (K\{ a}, ·) (called type II semifields). Let D be a semiring and d ɛ D. Then x ɛ S is said to be an additive identity of d iff x+d =d. The set of all additive identities of d in denoted by I [subscript D] (d). Theorem Let D be a ratio semiring and a a symbol not representing any element in D. Let S⊆ I [subscript D] (1) be such that either S = {u1D719} or S is an additive subsemigroup of I [subscript D] (1) such that D\S is an ideal of (D,+) if D is infinite. Then we can extend the binary operations of D K=D ∪ {a} making K into a semifield of type II such that (1) ax = xa =a for all x ɛ K, (2) a+x = x+a = a for all x ɛ S and a+x = x+a =1+x for all x ɛ D\S, (3) a+a = a or 1 if 1+1 =1, 1+1 ถ้า 1+1 ≠1. Theorem Let D be a ratio semiring and a symbol not representing any element in D. Let d ɛ D S⊆ I [subscript D] (d) be such that either S = Φ or S additive subsemigroup I [subscript D] (d) such that D\S is an ideal of (D,+) if D is infinite. Then we can extend the binary operations D to K=D ∪ {a} making K into a semifield of type III such that (1) ax = xa = dx for all x ɛ D and a² = d², (2) a+x = x+a = a for all x ɛ S amd a+x = x+a = d+x for all x ɛ D\S, (3) a+a = a or d if 1+1 = 1 , d+d if 1+1 ≠ 1. In this thesis we also study embedding theorems involving semirings, ratio semirings, semifields and fields. Let (K,+, ·) be a semifield. Then the prime semifield of K is the smallest semifield contained in K. In this thesis we show that prime semifields always exist and we determine all prime semifield of semifields.