Abstract
Bandelt and Petrich (1982) proved that an inversive semiring S is a subdirect product of a distributive lattice and a ring if and only if S satisfies certain conditions. The aim of this paper is to obtain a generalized version of this result. The main purpose of this paper however, is to investigate, what new necessary and sufficient conditions need we impose on an inversive semiring, so that, in its aforesaid representation as a subdirect product, the âringâ involved can be gradually enriched to a âfield.â Finally, we provide a construction of full Eâinversive semirings, which are subdirect products of a semilattice and a ring.
Highlights
In this paper, by a semiring we mean a nonempty set S together with two binary operations, â+â and â·â such that, (S, +) is a commutative semigroup and (S, ·) is a semigroup which are connected by ring-like distributivity
A subsemiring H of the direct product of two semiring S and T is called a subdirect product of S and T if the two projection mappings Ï1 : H â S given by Ï1(s, t) = s and Ï2 : H â T given by Ï2(s, t) = t are surjective
A semiring R which is isomorphic to a subdirect product H of S and T is called a subdirect product of S and T
Summary
By a semiring we mean a nonempty set S together with two binary operations, â+â and â·â (usually denoted by juxtaposition) such that, (S, +) is a commutative semigroup and (S, ·) is a semigroup which are connected by ring-like distributivity. An inversive semiring S is a subdirect product of a distributive lattice and a ring if and only if S satisfies the following conditions: A(1) a(a + a ) = a + a , A(2) a(b + b ) = (b + b )a, A(3) a + (a + a )b = a for all a, b â S, and A(4) a â S, a + b = b for some b â S implies a + a = a. An inversive semiring S is a subdirect product of an idempotent semiring and a ring if and only if S satisfies A(1) and A(4).
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