On certain semigroups of transformations that preserve double direction equivalence

Document Type : Research Paper

Author

School of Mathematical Science‎, ‎Guizhou Normal University‎, ‎Guiyang 550001‎, ‎China.

Abstract

Let TX be the full transformation semigroups on the set X. For an equivalence E on X, let TE*(X) = {α ∈ TX : ∀(x, y) ∈ E ⇔ (xα, yα) ∈ E} It is known that TE*(X) is a subsemigroup of TX. In this paper, we discuss the Green's *-relations, certain *-ideal and certain Rees quotient semigroup for TE*(X).

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L. Z. Deng, J. W. Zeng and B. Xu, Green's relations and regularity for semigroups of transformations that preserve double direction equivalence, Semigroup Forum 80 (2010), no. 3, 416--425.
J. B. Fountain, Abundant semigroups, Proc. London Math. Soc. (3) 44 (1982), no. 1, 103--129.
J. A. Green, On the structure of semigroups, Ann. of Math. (2) 54 (1951) 136--172.
H. S. Pei and H. J. Zhou, Abundant semigroups of transformations preserving an equivalence relation, Algebra Colloq. 18 (2011), no. 1, 77--82.
A. Umar, On certain infinite semigroups of order-decreasing transformations, Comm. Algebra 25 (1997), no. 9, 2987--2999.
A. Umar, On the semigroups of order-decreasing finite full transformations, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 1-2, 129--142.