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On the semigroup algebra of binary relations. (English) Zbl 1213.20064

Summary: The semigroup of binary relations on \(\{1,\dots,n\}\) with the relative product is isomorphic to the semigroup \(B_n\) of \(n\times n\) zero-one matrices with the Boolean matrix product. Over any field \(F\), we prove that the semigroup algebra \(FB_n\) contains an ideal \(K_n\) of dimension \((2^n-1)^2\), and we construct an explicit isomorphism of \(K_n\) with the matrix algebra \(M_{2^n-1}(F)\).

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
20M25 Semigroup rings, multiplicative semigroups of rings
15B34 Boolean and Hadamard matrices
03G15 Cylindric and polyadic algebras; relation algebras
16S36 Ordinary and skew polynomial rings and semigroup rings
20M30 Representation of semigroups; actions of semigroups on sets
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References:

[1] Clifton J. M., Proc. Amer. Math. Soc. 83 pp 248– (1981)
[2] DOI: 10.1090/S0002-9939-1977-0444823-9 · doi:10.1090/S0002-9939-1977-0444823-9
[3] Schwarz S., Czechoslovak Math. J. 20 pp 632– (1970)
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