On drl-semigroups and drl-semirings.

*(Russian. English summary)*Zbl 1373.06018Summary: In the article, drl-semirings are studied. The obtained results are true for drl-semigroups, because a drl-semigroup with zero multiplication is drl-semiring. These algebras are connected with the two problems: 1) there exists common abstraction which includes Boolean algebras and lattice ordered groups as special cases? [G. Birkhoff, Lattice theory. Rev. ed. New York: American Mathematical Society (AMS) (1948; Zbl 0033.10103)]; 2) consider lattice ordered semirings [L. Fuchs, Partially ordered algebraic systems. Oxford-London-New York-Paris: Pergamon Press (1963; Zbl 0137.02001)]. A possible construction obeying of the first problem is drl-semigroup, which was defined by K. L. N. Swamy [Math. Ann. 159, 105–114 (1965; Zbl 0135.04203)]. As a solution to the second problem, P. R. Rao introduced the concept of \(l\)-semiring in [Math. Semin. Notes, Kobe Univ. 9, 119–149 (1981; Zbl 0476.06018)]. We have proposed the name drl-semiring for this algebra.

In the present paper the drl-semiring is the main object. Results of Swamy [loc. cit.; Math. Ann. 160, 64–71 (1965; Zbl 0138.02104); ibid. 167, 71–74 (1966; Zbl 0158.02601)] for drl-semigroups are extended and are improved in some case. It is known that any drl-semiring is the direct sum \(S=L(S)\oplus R(S)\) of the positive to drl-semiring \(L(S)\) and l-ring \(R(S)\). We show the condition in which \(L(S)\) contains the least and greatest elements (Theorem 2). The necessary and sufficient conditions for the decomposition of a drl-semiring to a direct sum of an l-ring and a Brouwerian lattice (Boolean algebra) are founded at Theorem 3 (resp. Theorem 4). Theorems 5 and 6 characterize l-rings and cancellative drl-semirings by using symmetric difference. Finally, we proof that a congruence on a drl-semiring is a Bourne relation.

In the present paper the drl-semiring is the main object. Results of Swamy [loc. cit.; Math. Ann. 160, 64–71 (1965; Zbl 0138.02104); ibid. 167, 71–74 (1966; Zbl 0158.02601)] for drl-semigroups are extended and are improved in some case. It is known that any drl-semiring is the direct sum \(S=L(S)\oplus R(S)\) of the positive to drl-semiring \(L(S)\) and l-ring \(R(S)\). We show the condition in which \(L(S)\) contains the least and greatest elements (Theorem 2). The necessary and sufficient conditions for the decomposition of a drl-semiring to a direct sum of an l-ring and a Brouwerian lattice (Boolean algebra) are founded at Theorem 3 (resp. Theorem 4). Theorems 5 and 6 characterize l-rings and cancellative drl-semirings by using symmetric difference. Finally, we proof that a congruence on a drl-semiring is a Bourne relation.