Bulletin of the Korean Mathematical Society (대한수학회보)
- Volume 32 Issue 2
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- Pages.337-342
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- 1995
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- 1015-8634(pISSN)
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- 2234-3016(eISSN)
On spanning column rank of matrices over semirings
- Song, Seok-Zun (Department of Mathematics, Cheju National University, Cheju 690-756)
- Published : 1995.08.01
Abstract
A semiring is a binary system $(S, +, \times)$ such that (S, +) is an Abelian monoid (identity 0), (S,x) is a monoid (identity 1), $\times$ distributes over +, 0 $\times s s \times 0 = 0$ for all s in S, and $1 \neq 0$. Usually S denotes the system and $\times$ is denoted by juxtaposition. If $(S,\times)$ is Abelian, then S is commutative. Thus all rings are semirings. Some examples of semirings which occur in combinatorics are Boolean algebra of subsets of a finite set (with addition being union and multiplication being intersection) and the nonnegative integers (with usual arithmetic). The concepts of matrix theory are defined over a semiring as over a field. Recently a number of authors have studied various problems of semiring matrix theory. In particular, Minc [4] has written an encyclopedic work on nonnegative matrices.