ON drl-SEMIGROUPS AND drl-SEMIRINGS
https://doi.org/10.22405/2226-8383-2016-17-4-167-179
Abstract
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) ⊕ 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 an 6 characterize l-ring and cancellative drl-semiring by using symmetric difference. Finally, we proof that a congruence on drl-semiring is Bourne relation.
About the Author
O. V. Chermnykh
Vyatka State University
Russian Federation
Russian Federation
Leading teacher of the Chair of Fundamental and Computer Mathematics
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Review
For citations:
Chermnykh O.V. ON drl-SEMIGROUPS AND drl-SEMIRINGS. Chebyshevskii Sbornik. 2016;17(4):167-179. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-4-167-179
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