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Turkish Journal of Mathematics

DOI

10.55730/1300-0098.3496

Abstract

Let Xn denote the chain {1, 2, . . . , n} under its natural order. We denote the semigroups consisting of all order-preserving transformations and all orientation-preserving transformations on Xn by On and OPn , respectively. We denote by E(U) the set of all idempotents of a subset U of a semigroup S . In this paper, we first determine the cardinalities of Er(On) = {α ∈ E(On) : |im(α)| = |fix(α)| = r}, E ∗ r (On) = {α ∈ Er(On) : 1, n ∈ fix(α)}, Er(OPn) = {α ∈ E(OPn) : |fix(α)| = r}, E ∗ r (OPn) = {α ∈ Er(OPn) : n ∈ fix(α)} (1 ≤ r ≤ n) and then, by using these results, we determine the numbers of idempotents in On and OPn by a new method. Let OP− n denote the semigroup of all orientation-preserving and order-decreasing transformations on Xn . Moreover, we determine the cardinalities of OP− n , OP− n,Y = {α ∈ OP− n : fix(α) = Y } for any nonempty subset Y of Xn and OP− n,r = {α ∈ OP− n : |fix(α)| = r} for 1 ≤ r ≤ n. Also, we determine the number of idempotents in OP− n and the number of nilpotents in OP− n .

Keywords

order-decreasing transformation, Order-preserving transformation, orientation-preserving transformation

First Page

106

Last Page

117

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