# Algebraic and topological semirings

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This dissertation is concerned nrimarily with the extension to algebraic and topological semirings of the concepts of an inverse semigroup and divisibility in semigroups. Rather than follow the more usual and restrictive treatment of semirings employed in the earlier work in this area, we consider an algebraic semiring (S,+,.) as two algebraic semigroups, the additive structure (S,+) and the multiplicative semigroup (S,.) linked by a two-sided distributivity of multiplication over the addition. In Chapter 1 are introduced the basic definitions and terminology common to semiring theory. Semigroup concepts are generalized to both algebraic and topological semirings. A brief summary of well known results is presented and related material in nearrings and topological rings and topological groups is given for subsequent reference. Chapter 2 is concerned with algebraic semirings. A semiring (S,+,.) is said to be additively inverse provided that for each element x in S there exists a unique element x in S such that x = x + x + x and x = x + x + x. We shoxÂ«r that in such a semiring the product of any two elements under multiplication is contained in an additive group. If the semiring has a multiplicative identity, then addition is commutative and the additively inverse semiring is a union of additive groups. A characterization of additively inverse semirings, which are multiplicatively idempotent, is given as an additively commutative union of Boolean subrings. Chapter 2 is concluded with a necessary and sufficient condition that an additively commutative semiring be embeddable in a semiring which is a union of additive grouos. The embedding is possible if and only if the embedded semiring is additively separative (a + a = a + b = b + b implies that a = b). An element x of a semiring S is said to be additively divisible if for each positive integer n there exists an element y of S, depending upon x and n, such that x = ny. Chapter 3 is devoted to additively divisible semirings. In a compact semiring the set A of additively divisible elements contains the set E[+] of all additive idempotents. More importantly, the two complex products A.S and S.A are contained in E[+], Denoting the multiplicative idempotents by E[.], we obtain as an immediate corollary to this result the theorem of Selden that in a compact semiring S, which is the union of the sets S,E[.] and E[.].S, each additive subgroup is totally disconnected. The theorem of Anzai that xy = 0 for all elements x and y of a compact, connected topological ring follows as a second corollary. A characterization is then given for the multiplication of a semiring with an I-semigroup addition defined on an irreducible continuum. All such I-semigroup additions are divisible. Chapter 3 is concluded with results concerning general topological semirings. In Chapter U is presented a characterization of topological semirings, defined on an irreducible continuum, which are inversive in both their additive and multiplicative structures.