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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

ABSTRACT. A semi-g-cone is an algebra (C; * , : , · ) of type (2, 2, 2) satisfying the equations (a * a)). An -group cone is a semi-g-cone and a bounded semi-g-cone is term equivalent to a pseudo MV-algebra. Also, a subset A of a semi-g-cone C is an ideal of C if and only if it is a deductive system of its reduct (C; * , : ).

Structure theory for a generalised Boolean ringBy N.V. SUBRAHMANYAM in Waltair, IndiaTo Professor V. Ramaswami § 0. Introduction 0.0: Let R be any (associative) ring, in which PI: each element has a minimal idempotent duplicator in the centre of R; i.e. if a ~/t, there exists an idempotent a ° in the centre of R such that (i) aa ° = a, and (ii) for every idempotent e which satisfies ae = ea ---a, must aOe = a °.Such a ring R seems to be a natural generalisation for a Boolean ring (in terms of its lattice structure) and it includes all the known generalisations for {[3]) Boolean rings, the regular rings of vo~ N:~N~ with identity and without non-zero nilpotent elements etc., and further the so-called assoc~te rings of Ittv~G SUSSMA~ [3] which embrace all these classes of rings and also the biregular rings of AR~NS and KAPL/~NSK¥ etc. For these rings, the author has extended [2] the lattice theory [3] of SUSSMAN for associate rings and has also shown, by means of examples, that a Pl-ring (i,e. a ring with property P1) actually belongs to a wider class than the class of associate rings; and it was shown in [2] that a Pl-ring (and hence an associate ring) is composed of certain maximal sets, which are distributive lattices (Boolean rings), all isomorphic to one another: precisely, we have proved (see theorem 5 [2]) that each element of a Pl-ring is contained in a maximal set which is lattice isomorphic to the set R 0 of idempotents in it; and we were tempted to conjecture (which we shall disprove in this paper) to the effect that all the maximal sets are lattice isomorphic to one another (see Cor. 10.2 [2]). Further, a Pl-ring was shown to be isomorphic to a subdireet sum of subdireetly irreducible rings, each with unity element and (so) without other non-zero idempotent elements. 0.1: In the present paper, we examine the principal ideal structure of a Pl-ring satisfying the atomisti~ condition (see below for definition) and show that such a ring is isomorphic to a "special" subdirect sum of the ideals generated by its atomic idempotents, with further specialisations under additional conditions. These results extend those of IRVING Suss~z~ [4] for associate rings (to Pl.rings) which were obtained under the comp/ete a~m/~/c c~nd/tion and thus present improvements even in this case; also we shall show that many of our structure theorems cannot be improved. Finally, we obtain conditions on a Pl-ring which are necessary to render it an associate ring, and also present, in the last section, an example which seems to suggest certain directions in which the author's lattice theory for Prrings may be extended.Math. Ann. 141 21

ABSTRACT. We will show that the bricks (of Bosbach) and the pseudo MV-algebras are each term equivalent to the class of semigroups with a pair of unary operationsˆandˇsatisfying the equations: (âa)ˆb = b = b (aǎ)ˇand a(ba)ˇ= (bǎ)ˆb and also show that a brick is an interval [0, u] of the positive cone of a unital lattice ordered group. We further extend the notion of implications to a pseudo MV-algebra and study the algebra of such implications.