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On $drl$-semigroups and $drl$-semirings
O. V. Chermnykh Vyatka State University
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
$S=L(S)\oplus R(S)$ 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.
Keywords:
semiring, $drl$-semigroup, $drl$-semiring, lattice ordered ring.
UDC:
512.558 Received: 11.05.2016
Accepted: 13.12.2016
DOI:
10.22405/2226-8383-2016-17-4-167-179
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