Boolean semirings

Abstract

IntroductionA ring R, in which every element is idempotent, is called a "Boolean ring"; and it is well-known that a Boolean ring (with unity element) is also a distributive (and complemented) lattice.Several generalisations for a Boolean ring are known --p-rings, regular rings, biregular rings, associate rings [3], Pl-rings [I], etc., of which, the class of associate rings contains, as special cases, those of p-rings, regular rings with unity element and without nonzero nilpotent elements etc. [3], and the class of Pl-rings contains those of associate rings and also biregular rings [1].Recently, we have shown [1], extending SUSSMA~'S [3] results of the same kind for associate rings, that a Pl-ring admits a natural extension of the Boolean lattice theory, and is composed (= logical union) of certain ,,maximal" sets, which are Boolean rings (and hence, also distributive lattices), all isomorphic to one another. Further, the structure theorems of Boolean rings have been extended (see [2] and also [4]) to Pl-rings, where we have studied the structure of a Pl-ring in terms of its "idem-ideals" (=principal ideals generated by idempotent elements) under various conditions: atomlsticity, complete atomisticity, etc. Also, all these results are valid for a more general situation than that of Pl-rings, as was indicated in [2]: see § 5, page 309.This present study was prompted by the fact that a Pl-ring has more "lattice structure" than was exhibited in [i] or [3]; in fact, it is almost a distributive lattice (see theorem 3 and its corollary below). This structure was suspected to be related, in some stronger way, to the concept of a Boolean ring than the one we have chosen to define a Pl-ring (and hence, also for the other ring-generalisations for a Boolean ring); and this is indeed the case, as will be shown in the present paper.We introduce the notion of a "Boolean semiring" (see the definition 1 below), which differs from the Boolean ring, only to the extent that one of the distributive laws is replaced by the identity: abe = bac; and it is shown that a PI" ring can be regarded as a Boolean semiring, and a preliminary investigation of the "lattice" structure of a Boolean semiring is made. Finally, the concept of a Boolean semiring, as presented here, is natural to tempt one at further generalisations of a Boolean semiring along the lines employed earlier in the

Medial near-rings

“…Left permutable near-rings which also satisfy x 2 = x identically are the "t-near-rings" of LIGI4 [18] and have been considered by RATLIFF [27] and SUBRAHMAYAN [29]. (The latter calls them "Boolean semirings".)…”

Left self distributive near-rings

“…PROPOSITION Boolean, left permutable near-rings were called "^-near-rings" by Ligh who characterized them in terms of subdirectly irreducible components [ 15]. (Earlier Subrahmanyan [28] considered Boolean, left permutable near-rings with abelian additive group, which he called "Boolean semirings.") It is immediate that a /Nnear-ring is LSD.…”

The structure of a special class of near-rings

“…Since special Boolean near-rings and Boolean semi-rings as defined in [3] and [8] respectively are ^-near-rings, Theorem 5.2 furnishes the subdirect structures of those near-rings as well.…”