On $drl$-semigroups and $drl$-semirings

O. V. Chermnykh

Čebyševskij sbornik, Tome 17 (2016) no. 4, pp. 167-179

Résumé

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. This 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); 2) consider lattice ordered semirings (L. Fuchs). A possible construction obeying of the first problem is $drl$-semigroup, which was defined by K. L. N. Swamy in 1965. As a solution to the second problem, Rango Rao introduced the concept of $l$-semiring in 1981. We have proposed the name $drl$-semiring for this algebra. In the present paper the $drl$-semiring is the main object. Results of K. L. N. Swamy 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 of decomposition of $drl$-semiring to direct sum of $l$-ring and Brouwerian lattice (Boolean algebra) are founded at theorem 3 (resp. theorem 4). Theorems 5 and 6 characterize $l$-ring and cancellative $drl$-semiring by using symmetric difference. Finally, we proof that a congruence on $drl$-semiring is Bourne relation. Bibliography: 11 titles.

Détail

Keywords: semiring, $drl$-semigroup, $drl$-semiring, lattice ordered ring.