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Algorithm for Computing J Relations in the Monoid of Boolean Matrices  

Han, Jae-Il (국민대학교 전자정보통신대학 컴퓨터공학부)
Publication Information
Journal of Information Technology Services / v.7, no.4, 2008 , pp. 221-230 More about this Journal
Abstract
Green's relations are five equivalence relations that characterize the elements of a semigroup in terms of the principal ideals. The J relation is one of Green's relations. Although there are known algorithms that can compute Green relations, they are not useful for finding all J relations in the semigroup of all $n{\times}n$ Boolean matrices. Its computation requires multiplication of three Boolean matrices for each of all possible triples of $n{\times}n$ Boolean matrices. The size of the semigroup of all $n{\times}n$ Boolean matrices grows exponentially as n increases. It is easy to see that it involves exponential time complexity. The computation of J relations over the $5{\times}5$ Boolean matrix is left an unsolved problem. The paper shows theorems that can reduce the computation time, discusses an algorithm for efficient J relation computation whose design reflects those theorems and gives its execution results.
Keywords
Semigroup; Ideal; Green Relation; Boolean matrix; Algorithm; Computational Complexity;
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Times Cited By KSCI : 3  (Citation Analysis)
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