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Semigroups and their natural order. (English) Zbl 0816.20057
For any semigroup \(S\) the relation defined by \(a \leq b\) if and only if \(a = xb = by\), \(xa = a\) for some \(x,y \in S^ 1\) is called the natural partial order on \(S\) and is known to generalize the well-known relation of the same name for regular semigroups. The paper addresses and partially answers the general questions: When is \(\leq\) trivial? When is it total? When is it compatible with multiplication? Applications are given concerning least primitive congruences, retract extensions of regular semigroups, and strong semilattices of semigroups of certain types. Along the way, an extension of the corollary of Green’s Lemma on the equipotency of \(\mathcal H\)-classes within a \(\mathcal D\)-class is proved, extending the result to the corresponding order ideals of such elements.

MSC:
20M10 General structure theory for semigroups
06F05 Ordered semigroups and monoids
20M15 Mappings of semigroups
20M17 Regular semigroups
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References:
[1] BLYTH T., GOMES G.: On the compatibility of the natural partial order on a regular semigroup. Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), 79-84. · Zbl 0511.20043
[2] CLIFFORD A. H., PRESTON G. B.: The Algebraic Theory of Semigroups. Vol. I, II. Math. Surveys Monographs 7, Amer. Math. Soc, Providence, R.I., 1961, 1967. · Zbl 0111.03403
[3] FARÉS. N.: Idempotents et D-classes dans les demigroupes et les anneaux. C.R. Acad. Sci. Paris 269 (1969), 341-343. · Zbl 0186.31203
[4] GREEN J. A.: On the structure of semigroups. Ann. of Math. 54 (1951), 163-172. · Zbl 0043.25601
[5] HARTWIG R.: How to partially order regular elements. Math. Japon. 25 (1980), 1-13. · Zbl 0442.06006
[6] HICKEY J.: On variants of a semigroup. Bull. Austral. Math. Soc. 34 (1986), 447-459. · Zbl 0586.20030
[7] HIGGINS P.: Techniques of Semigroup Theory. Clarendon Press, Oxford, 1992. · Zbl 0744.20046
[8] MITSCH H.: A natural partial order for semigroups. Proc. Amer. Math. Soc. 97 (1986), 384-388. · Zbl 0596.06015
[9] MITSCH H.: Subdirect products of E-inversive semigroups. J. Austral. Math. Soc. Ser. A 48 (1990), 66-78. · Zbl 0691.20050
[10] NAMBOORIPAD K. S.: The natural partial order on a regular semigroup. Proc. Edinburgh Math. Soc. (2) 23 (1980), 249-260. · Zbl 0459.20054
[11] PETRICH M.: On extensions of semigroups determined by partial homomorphisms. Indag. Math. 28 (1966), 49-51. · Zbl 0136.26601
[12] PETRICH M.: Congruences on strong semilattices of regular, simple semigroups. Semigroup Forum 37 (1988), 167-199. · Zbl 0643.20041
[13] PETRICH M.: Introduction to Semigroups. Ch. E. Merill Publishing Company. Columbus, Ohio, 1973. · Zbl 0321.20037
[14] VÁGNER V. V.: Generalized groups. (Russian), Dokl. Akad. Nauk. SSSR (N.S.) 84 (1952), 1119-1122.
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