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Left orders in completely \(0\)-simple semigroups. (English) Zbl 0855.20048

An element \(a\) of a semigroup \(S\) is said to be square-cancellable if, for all \(x,y\in S^1\), \(a^2x=a^2y\) implies \(ax=ay\) and \(xa^2=ya^2\) implies \(xa=ya\). Let \(S\) be a subsemigroup of \(Q\). Then \(Q\) is a semigroup of left quotients of \(S\) and \(S\) is a left order in \(Q\) if every square-cancellable element of \(S\) lies in a subgroup of \(Q\) and every \(q\in Q\) can be written as \(q=a^\# b\) where \(a\) is square-cancellable, \(b\in S\), and \(a^\#\) is the inverse of \(a\) in a subgroup of \(S\) which contains \(a\) [see J. Fountain, M. Petrich, J. Algebra 101, 365-402 (1986; Zbl 0589.20041)]. A semigroup \(S\) with zero is called prime if for any \(a\neq 0\neq b\) in \(S\) there is an \(x\in S\) with \(axb\neq 0\). Let \(\lambda\) and \(\rho\) be binary relations on \(S\) such that \(a\lambda b\) iff \(Sa\cap Sb\neq 0\), and \(a\rho b\) iff \(aS\cap bS\neq 0\). Denote by \(\lambda^t\) and \(\rho^t\) the transitive closure of \(\lambda\) and \(\rho\), respectively. The main result says that a semigroup \(S\) with zero is a left order in a completely 0-simple semigroup iff (i) \(S\) is prime and \(S^1\) categorical at zero, (ii) there exists an \(a\in S\) such that \(aSa\) is a left order in a group with zero, (iii) if \(x(\lambda^t\cap\rho^t)y\) in \(S\) then, for any \(z\in S\), \(zx=zy\neq 0\) implies \(x=y\) and \(xz=yz\neq 0\) implies \(x=y\). This theorem is used to obtain several alternative characterizations of left orders in completely 0-simple semigroups.

MSC:

20M10 General structure theory for semigroups

Citations:

Zbl 0589.20041
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References:

[1] Ánh, P. N., V. Gould, and L. Márki,Completely 0-simple semigroups of left quotients of a semigroups, Internat. J. Algebra Comput. (to appear). · Zbl 0851.20054
[2] Fountain, J., and V. A. R. Gould,Completely 0-simple semigroups of quotients II, in ”Contributions to General Algebra 3” (Proc. Conf. Wien, 1984), pp. 115–124. Verlag Hölder-Pichler-Tempsky, Wien, 1985.
[3] Fountain, J., and M. Petrich,Completely 0-simple semigroups of quotients, J. Algebra101 (1986), 365–402. · Zbl 0589.20041 · doi:10.1016/0021-8693(86)90200-0
[4] Fountain, J., and M. Petrich,Completely 0-simple semigroups of quotients III, Math. Proc. Cambridge. Phil. Soc.105 (1989), 263–275. · Zbl 0687.20056 · doi:10.1017/S030500410006775X
[5] Gould, V. A. R.,Left orders in regular H-semigroups II, Glasgow Math. J.32 (1990), 95–108. · Zbl 0692.20049 · doi:10.1017/S0017089500009101
[6] Hotzel, E.,Dual D-operands and the Rees theorem, in ”Algebraic Theory of Semigroups” (Proc. Conf. Szeged, 1976), pp. 247–275. Colloq. Math. Soc. J. Bolyai,20, North-Holland, Amsterdam, 1979.
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