Abstract
A partial semiring is a structure possessing an infinitary partial addition and a binary multiplication, subject to a set of axioms. The partial functions under disjoint-domain sums and functional composition is a partial semiring. In this paper we introduce the notions of ( R, S ) - partial bi-semimodule and ( R, S ) - homomorphism of ( R, S ) - partial bi-semimodules and extended the results on partial semimodules over partial semirings by P. V. Srinivasa Rao (8) to ( R, S ) - partial bi-semimodules.
Highlights
Defined infinitary operations occur in the contexts ranging from integration theory to programming lanugage semantics
The general cardinal algebras studied by Tarski in 1949, ∑- structures studied by Higgs in 1980, Housdorff topoligical commutative groups studied by Bourbaki in 1966, sum-ordered partial monoids and sum-ordered partial semirings studied by Arbib, Manes, Benson and Streenstrup are some of the algebraic structures of the above type
In this paper we introduce the notions of ( R, S ) - partial bi-semimodule, ( R, S ) - homomorphism and absorbing subbi-semimodules and we generalise the results of semirings
Summary
Defined infinitary operations occur in the contexts ranging from integration theory to programming lanugage semantics. The study of pfn(D, D) (the set of all partial functions of a set D to itself), Mfn(D, D) (the set of all multi functions of a set D to itself) and Mset(D, D) (the set of all total functions of a set D to the set of all finite multi sets of D ) play an important role in the theory of computer science, and to abstract these structures Manes and Benson[6] introduced the notion of sum ordered partial semirings(so-rings). Motivated by the work done in partially-additive semantics by Arbib, Manes[3] and in the development of matrix theory of so-rings by Martha E. V. Srinivasa Rao[9] in 2011 developed the ideal theory for so-rings and partial semimodules over partial semirings. P.V [9]) to the class of (R, S ) – partial bi-semimodules
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