J. Nig. Soc. Phys. Sci. 4 (2023) 878 Journal of the Nigerian Society of Physical Sciences On Nonexpansive and Expansive Semigroup of Order-Preserving Total Mappings in Waist Metric Spaces I. F. Usamota, O. T. Wahabb,∗, S. M. Alatac, K. R. Tijanid a Department of Mathematics, University of Ilorin, Ilorin, Nigeria bDepartment of Mathematics and Statistics, Kwara State University, Malete, Nigeria c Department of Computer Science, College of Arabic and Islamic Legal Studies, Ilorin, Nigeria dDepartment of Mathematical Sciences, Osun State University, Osogbo, Nigeria Abstract In this paper, we introduce nonexpansive and expansive semigroup of order-preserving total mappings (ONTn) and (OETn), respectively, to prove some fixed point theorems in waist metric spaces. We examine the existence of mappings that satisfy the conditions ONTn and OETn. We also prove that every semigroup of order-preserving total mappings OTn has fixed point properties and that the set of fixed points is closed and convex. The present study generalised many previous results on semigroup of order-preserving total mappings OTn. Efficacy of the results was justified with some practical examples. DOI:10.46481/jnsps.2022.878 Keywords: Fixed point, semitopological semigroup, order-preserving total mappings, waist metric space, nonexpansive map. Article History : Received: 20 June 2022 Received in revised form: 08 November 2022 Accepted for publication: 09 November 2022 Published: 22 December 2022 c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: Joel Ndam 1. Introduction In the last four decades, semigroup of mappings is one of the areas of application in fixed point theory. In algebra, a semi- group is simply a set S with an associative binary operation. A subset P ⊂ S is called a subsemigroup of S if it is closed under the binary operation on S . Let Xn be an ordered finite set in a standard way and let α : Dom(α) ⊆ Xn → Xn be a self-map. The map α is called a full or total transformation of Xn if Dom(α) = Xn. It is said to be partial if Dom(α) ⊆ Xn. Otherwise, it is called partial one-to-one or strictly partial. The set of full transformations on ∗Corresponding author tel. no: +2348035224754. Email address: taofeek.wahab@kwasu.edu.ng ( O. T. Wahab ) Xn, denoted by Tn, forms a semigroup under the composition of mappings called the full transformation semigroup. The set of partial and partial one-to-one transformations on Xn, denoted by Pn and In, respectively, also form a semigroup under the composition of mappings. The semigroup of order-preserving full transformation of Xn is defined by OTn = {α ∈Tn : x ≤ y ⇒ xα ≤ yα, for all x, y ∈ Xn} . Let α be a transformation in OTn. A point x∗ ∈ Xn is said to be fixed if it coincides with the image α. For α ∈ OTn, the fix of α is given by Fix(α) = {x ∈ Xn : xα = x}. Let OCTn and OC∗Tn be subsemigroups of OTn, then a map- 1 Usamot et al. / J. Nig. Soc. Phys. Sci. 4 (2023) 878 2 ping α ∈ OCTn is called a ’contraction’ if |xα− yα| ≤ |x − y| for all x, y ∈ Xn (1) and a mapping α ∈ OC∗Tn is called ’contractive’ if |xα− yα| ≥ |x − y| for all x, y ∈ Xn (2) It is worthy to note there is a difference between ’contrac- tion’ and ’contractive’ transformations in semigroup and de- terministic fixed points. We refer to the following references for standard concepts and terminologies in the semigroup, (see [1, 2, 3, 4, 5]). If α preserves only distance (with no order), then it satisfies both (1) and (2). This is called an isometry semigroup, |xα− yα| = |x − y| for all x, y ∈ Xn. (3) Several fixed point results have been proved on semigroup for the family of isometries under the asymptotic nonexpan- sive operators [6, 7, 8, 9], Lipschitzian semigroup of mappings [10, 11], commutative semigroup [12] etc. Worthy to mention a few recent studies of fixed points when the parameter set of semigroups is equal to {0, 1, 2, 3, . . .} and Tn = T n is the n- th iterate of asymptotic pointwise contractions and asymptotic nonexpansive mappings in metric spaces (see [9, 13]). Also, a procedure for constructing and finding the cardinality of order- preserving total transformations with finite fixed points have been considered in [14, 15]. However, we observed through a survey that few or no record of results concerning the exis- tence of the fixed points of a semigroup of order-preserving to- tal mappings. In this respect, existence of semigroup of order- preserving full transformation (which double as a nonlinear op- erator on Xn) is studied in the present paper. The intuitive notion of semigroup is integrated into a more robust geometric struc- ture to unify some results in the semigroup theory. In concrete, the paper introduces some fixed point theorems for the non- expansive (and expansive) semigroup of order-preserving map- pings to prove some existence of fixed points of the elements of subsemigroups OCTn (and OC∗Tn). We recall Banach’s contraction mapping principle [16] which has been used in many areas of applied sciences to study the existence properties of nonlinear operators. Definition 1. Let (E, d) be a metric space. A map T : E → E is called contraction on E if there exists a constant λ ∈ [0, 1) such that for all x, y ∈ E, d(T x, T y) ≤ λd(x, y) (4) If the condition (4) is weakened, that is λ = 1, then it reduces to a nonexpansive mapping d(T x, T y) ≤ d(x, y) (5) Otherwise, it is an expansive mapping. We note that every map- ping T ∈ OCTn is a nonexpansive mapping and every mapping T ∈ OC∗Tn is an expansive mapping. The inclusion in both cases are strict. For few older results on the family of nonex- pansive mappings, see [6, 17, 18, 19, 20, 21]. 2. Waist Metric Space Let α : Dom(α) → Im(α) be a map in OTn, where Dom(α), Im(α) ⊂ X. The right waist and left waist of Dom(α) are given, respectively, by w+(Dom(α)) = max {|x| : x ∈ Dom(α)} and w−(Dom(α)) = min {|x| : x ∈ Dom(α)} . Similarly, the right and left waist of Im(α) are, respectively, given by w+(α) = max {|y| : y ∈ Im(α)} and w−(α) = min {|y| : y ∈ Im(α)} . In view of the above, we introduce a notion of distance func- tion with the left waist ω−(·, ·) and right waist ω+(·, ·) terms as follow: Definition 2. Let M be a non-empty ordered set and X be a finite subset of M. A function ω : X × X → X ∪{0} is called a right and left waist metric if for given transformation α and for each x, y ∈ Dom(α) ⊆ X, the following conditions hold: W1: ω+(x, y) and ω−(x, y) are finite and nonnegative integer; W2: ω+(x, x) = 0 and ω−(x, x) = 0; W3: ω+(x, y) = ω+(y, x) and ω−(x, y) = ω−(y, x); W4: ω+(x, y) + ω+(y, z) ≥ ω+(x, z) and ω−(x, y) + ω−(y, z) ≥ ω−(x, z) for x, y, z ∈ X. The pair (M,ω)α is called a waist metric space (WMS). WMS is a weakening form of the canonical metric space and it is clas- sified as pseudometric space. Example 1. Let X = {1, 2} ⊂ M be endowed with the waist distance ω+X (x, y) = max {|x − y| : x, y ∈ X} and α = (1)(2), then ω + X (x, y) is a waist metric on X. Similarly for ω−X (x, y). Example 2. Let α be a total map on set X = {1, 2, 3, 4, 5} ⊂ M such that α = (1)(4)(2 1](3 4](5 4] ∈ OTn, observe that Im(α) = {1, 4} and Dom(α) = {1, 2, 3, 4, 5}. The following are verifiable: i w+(Dom(α)) = 5 and w−(Dom(α)) = 1. ii Both ω+X (x, y) and ω − X (x, y) are waist metric on X. Remark 1. If α ∈ OTn for any given set X, then ω−(x, y) = ω+(x, y). On the other hand, this is not so if α is a partial map PTn. Since the main focus of this present study is on the map- pings in OTn, we denote ω(x, y) by a waist metric with no em- phasis on left or right waist metric. 2 Usamot et al. / J. Nig. Soc. Phys. Sci. 4 (2023) 878 3 2.1. Completeness of (M,ω)α Let {xk} be a sequence in X ⊂ (M,ω)α. Since X is a finite set, the convergent of {xk} is vacuously satisfied. We present the following useful lemmas. Lemma 1. A finite set X ⊂ M is a closed set. Proof: Let X = {x1, x2, . . . , xn} be a finite set and let X∪X′ = M, where X ∩ X′ = ∅. For each xi ∈ M, there is an ε−net such that x j ∈ B(xi,ε) ⊆ X′ for i , j. Observe that each B(xi,ε) is an open ball in M. Let X′ = ∪i∈∆ {B(xi,ε)}, then X′ is the union of open balls which itself is an open set in M. Now, M\X′ = ∩i∈∆ {M\B(xi,ε)} = {x1, x2, . . . , xn}. That is, X = M\X′ is a complement of an open set. Thus, X is closed. Remark 2. In Lemma 1, observe that for each y ∈ X, B(x,ε) ∩ X = {x}. This means that no point in X is an accumulation point but every point in X is an isolated point. More so, any metric on a finite space induces a discrete topology (see [22]). Definition 3. Let (M,ω)α be a WMS and X ⊂ M. A sequence xk ∈ X is said to be a Cauchy sequence in X if for given ε−net, there exist l and k with l ≥ k such that xl ∈ B(xk,ε). Lemma 2. Any convergent sequence in any metric space is a Cauchy sequence. Definition 4. A waist metric space (M,ω)α is said to be com- plete if every Cauchy sequence in M converges to an element in M. Theorem 1. Let (M,ω)α be a complete waist metric space and X ⊂ M. The subspace (X,ω)α is complete if and only if X is a closed subset of M. The proof follows from Lemma 1 and 2. The following con- cepts are versions of some results in [23, 24]. Definition 5. Let (M,ω)α be a waist metric space. A mapping u : M × M ×[0, 1] → M is called a convex structure on M if for all x, y ∈ M and λ ∈ [0, 1] ω(z, u(x, y,λ)) ≤ λω(z, x) + (1 −λ)ω(z, y) holds for all z ∈ M. The waist metric space (M,ω)α together with a convex structure uλ = u(x, y,λ) is called a convex waist metric space. In Definition 5, a convex waist metric space (M,ω, u)α sat- isfies the following: ω(u(x, p,λ), u(y, p,λ)) ≤ λω(x, y), x, y, p ∈ M, λ ∈ [0, 1] ω(x, y) = ω(x, u(x, p,λ)) + ω(u(y, p,λ), y), x, y ∈ M, λ ∈ [0, 1] Definition 6. A nonempty subset X of a convex waist metric space (M,ω, u)α is said to be convex if u(x, y,λ) ∈ X for all x, y ∈ X and λ ∈ [0, 1]. Definition 7. A nonempty subset X is said to be p-starshaped, where p ∈ X, provided u(x, p,λ) ∈ X for all x ∈ X and λ ∈ [0, 1], that is, the segment [ p, x] = {u(x, p,λ) : 0 ≤ λ ≤ 1} joining p to x is contained in X for all x ∈ X. The set X is said to be starshaped if it is p-starshaped for some p ∈ X. Clearly, each convex waist metric space is starshaped but not conversely. Lemma 3. Let (M,ω)α be a waist metric space. Then ω2(z, u 1 2 ) ≤ 1 2 ω2(z, x) + 1 2 ω2(z, y) − 1 4 ω2(x, y) (6) for all x, y, z ∈ M. The proof follows from the parallelogram law. Inequality (6) is similar to the (CN) inequality of Bruhat and Tits [26]. 2.2. Nonexpansive Semigroup of Order-preserving Maps Let S be a semitopological semigroup and X be a nonempty closed subset of a waist metric space (M,ω)α. A family ℘ = {αs : s ∈ S} of mappings of X into itself is called a semigroup if it satisfies the following: S1: xαst = xαsαt for all s, t ∈ S and x ∈ X; S2: for every x ∈ X the mapping s → αs x from S into X is continuous. The set of all fixed points of semigroup mappings is denoted by F(αs) = {x ∈ X : xαs = x for s ∈ S}. More so, any map αs ∈ OCTn or αs ∈ OC ∗Tn possesses the properties as stated in section one. Note that (i) the set of fixed points of order-preserving maps, denoted by Fo(αs ), is a subset of F(αs). (ii) Fo(αs ) ⊂ F n o (αs ) for all n > 1. (iii) If F n o (αs ) is singleton for some n, so does Fo(αs ). Without loss of generality, the notion ’nonexpansive’ connotes ’contraction’ while ’expansive’ connotes ’contractive’. In view of the above, we present some definitions of nonexpansive semigroup of order-preserving mappings in WMS under the property that both OCTn and OC∗Tn have no common maps. Definition 8. Let X be a closed subset of (M,ω)α and let S be a semitopological semigroup. The map αs : X → X is called a nonexpansive semigroup of order-preserving total map ONTn if for x, y ∈ X and s ∈ S , ω(xαs, yαs ) ≤ ω(x, y). (7) Definition 9. Let X be a closed subset of (M,ω)α and let S be a semitopological semigroup. The map αs : X → X is called an expansive semigroup of order-preserving total map OETn if for x, y ∈ X and s ∈ S , ω(xαs, yαs ) > ω(x, y). (8) 3 Usamot et al. / J. Nig. Soc. Phys. Sci. 4 (2023) 878 4 3. Main Results The following lemma is useful in the proof of the main re- sults. Lemma 4. If ℘ = {αs : s ∈ S} is a semigroup of continuous mappings of X into itself and ω(αs x, y) → 0 as s → ∞R for x, y ∈ X, then y ∈ Fo(αs) ⊂ X. Proof: Let ε > 0 be given. By the continuity of αt for t ∈ S , there exists τ > 0 such that ω(αt x,αty) < ε 2 whenever ω(x, y) < τ for x, y ∈ X. Also, since ω(αs x, y) → 0 as s → ∞R, then there exists u ∈ S such that ω(αau x, y) < min{ ε 2 ,τ} for each a ∈ S . Thus, ω(αtau x,αty) < ε 2 . Now, ω(y,αty) ≤ ω(y,αtau x) + ω(αtau x,αty) < ε 2 + ε 2 = ε Since ε > 0 is arbitrary, it follows that y ∈ Fo(αs). Remark 3. If S = N, then the hypothesis on αs would include asymptotic regularity condition. For example, see Theorem 6.7. in [25]. In the next theorem, we let A(X, c̄o{αs x}) denote the asymptotic center and r(x0, c̄o{αs x} is the asymptotic radius. Theorem 2. Let X be a nonempty closed subset of a convex waist metric space (M,ω)α and let S be a semitopological semi- group. Suppose αs ∈ ℘ is a nonexpansive semigroup of order- preserving total mapping (7) of X into itself, that is, αs ∈ ONTn for s ∈ S . If the set {αs x, s ∈ S} is bounded for some x ∈ X and y ∈ A(X, c̄o{αs x}), then y ∈ Fo(αs). Proof: Let {αs x, s ∈ S} be a bounded net. Define R := r(y, c̄o{αs x}) for y ∈ A(X, c̄o{αs x}) with the property that ω(x, y) < R. If R = 0, then lim sup ω(αs x, y) = 0 and by Lemma 4, the proof is complete. On the other hand, suppose R > 0 and y < Fo(αs), then for given ε > 0 and a subnet {sβ} in S , we have ω(αsβ y, y) > ε, for sβ ∈ S. Also, since ω(αs x, y) → 0 as s → ∞R, then there exists γ ∈ S such that, for choosing ν ≥ 0, ω(αγ x, y) < R + ν. (9) Moreover, we have by hypothesis that ω(αs x,αsy) ≤ lim sup β ω(αβx,αβy) + ν ≤ lim sup ω(x, y) + ν = R + ν (10) By Lemma 3, (9) and (10), we have ω2(u,αs x) ≤ 1 2 ω2(u,αs x) + 1 2 ω2(αs x,αsy) − 1 4 ω2(y,αsy) ≤ 1 2 (R + ν)2 + 1 2 (R + ν)2 − 1 4 ε2 ≤ (R −ν)2 Thus, ω(u,αs x) < R −ν which implies that r(u,αs x) < r(y, c̄o{αs x}). is a contradiction. Hence, y ∈ Fo(αs). Necessary and sufficient result for Theorem 2 is presented as follow: Theorem 3. Let X be a nonempty closed subset of a convex waist metric space (M,ω)α and let S be a semitopological semi- group. Suppose αs ∈ ℘ is a nonexpansive semigroup of order- preserving total mapping (7) of X into itself, that is, αs ∈ ONTn for s ∈ S . The set {αs x, s ∈ S} is bounded for some x ∈ X if and only if Fo(αs) is nonempty. Proof: Assume that {αs x, s ∈ S} is bounded for some x ∈ X, there is a unique element y ∈ X for which y ∈ A(X, c̄o{αs x}). By Theorem 2, Fo(αs) is nonempty. The converse is obvious. Remark 4. If in Theorem 2, the boundedness assumption on {αs x, s ∈ S} is dropped, then another suitable concept is stated in the next theorem. Theorem 4. Let X be a closed subset of a complete waist metric space (M,ω)α and let S be a semitopological semigroup. Sup- pose αs ∈ ℘ is a nonexpansive semigroup of order-preserving total mapping (7) of X into itself, that is, αs ∈ ONTn for s ∈ S . Then, αs has at least one fixed point. Proof: For δ ∈ (0, 1], set Tδ = (1 − δ)αs. It follows that Tδ is a δ-contraction on X and by the Banach fixed point theorem, there exists xδ for δ ∈ (0, 1] such that Tδxδ = xδ. Now, we show that for δk → 0, the net xδk converges to p, where p is a fixed point of αs. Indeed, for any arbitrary u ∈ X, we have ω2(xδk, u) = ω 2(xδk − p, u − p) = ω2(xδk, p) + ω 2( p, u) + 2ω(xδk − p, u − p) ≤ ω2(xδk, p) + ω 2( p, u) By setting u = αs p, we have lim sup ( ω2(xδk,αs p) −ω 2(xδk, p) ) ≤ ω2( p,αs p) (11) Also, since Tδk xδk = xδk and δk → 0 as k →∞, then lim k→∞ ω(xδk,αs xδk ) = lim k→∞ [ω(xδk, Tδk xδk ) + δkω(0,αs xδk )] = lim k→∞ ω(xδk, Tδk xδk ) → 0 (12) On the other hand, since αs ∈ ONTn, then ω(αs p,αs xδk ) ≤ ω(p, xδk ) (13) We have from (12) and (13) that ω(xδk,αs p) ≤ ω(xδk,αs xδk ) + ω(xδk, p) which further implies lim sup[ω(xδk,αs p) −ω(xδk, p)] ≤ lim k→∞ ω(xδk,αs xδk ) = 0 (14) 4 Usamot et al. / J. Nig. Soc. Phys. Sci. 4 (2023) 878 5 Also, from (11) and (14), we obtain lim sup ( ω2(xδk,αs p) −ω 2(xδk, p) ) = lim sup ( ω(xδk,αs p) −ω(xδk, p) ) × ( ω(xδk,αs p) + ω(xδk, p) ) Thus, lim sup ( ω2(xδk,αs p) −ω 2(xδk, p) ) = ω2(p,αs p) = 0 and hence, p is the fixed point of αs Theorem 5. Let X be a closed subset of a complete waist met- ric space (M,ω)α and let S be a semitopological semigroup. Suppose αs ∈ ℘ is a mapping satisfying (7), that is, αs ∈ ONTn for s ∈ S . Then, the set Fo(αs) ⊂ X is a nonempty closed convex set. Proof: Since αs satisfies (7), by Theorem 2, αs has fixed point in X. It is left to show that Fo(αs) is closed and convex. Firstly, we show that Fo(αs) is closed. Let {xt} be a net in Fo(αs) such that xt → x, then by hypothesis: ω(αt x, x) ≤ ω(αt x, xt) + ω(xt, x) → 0 This implies αt x → x ∈ Fo(αs). Hence, Fo(αs) is closed. Also, let x, y ∈ Fo(αs) and λ ∈ [0, 1], we have uλ = u(x, y,λ) ∈ Fo(αs). Indeed, ω(αsuλ, x) = ω(αsuλ,αs x) ≤ ω(uλ, x) Similarly, ω(αsuλ, y) ≤ ω(uλ, y). Thus, ω(x, y) ≤ ω(x,αsuλ) + ω(αsuλ, y) ≤ ω(x, y). This shows that for some a, b with 0 ≤ a, b ≤ 1, we have ω(x,αsuλ) = aω(x, uλ) and ω(y,αsuλ) = bω(y, uλ) from which it follows that αsuλ ∈ Fo(αs). By letting S = N in Theorems 4 and 5, we obtain the existence property of a nonexpansive semigroup of order-preserving total mapping in waist metric spaces as follow: Corollary 1. Let X be a closed subset of a convex waist metric space (M,ω). Suppose α ∈ ℘ = {αn : n ∈ N} is a nonexpansive semigroup of order-preserving total mapping (7) of X into itself. Then. Fo(αs) , ∅ if and only if {αn x : n ∈ N} is bounded for some x ∈ X. Furthermore, Fo(αs) is closed and convex. Theorem 6. Let X be a closed subset of a complete waist met- ric space (M,ω)α and let S be a semitopological semigroup. Suppose αs ∈ ℘ = {αs : s ∈ S} is a mapping satisfying (8). Then, i. the map αs exists; ii. Fo(αs) is nonempty. Proof: i. Let x1, x2 ∈ X with x1 < x2 . Suppose, on contrary, that αs < OETn, then αs is an order-reversing map, that is, x1αs ≥ x2αs. By induction, xkαs ≥ xk+1αs, for k = 1, 2, 3, . . . , n − 1 Define xk+1 = xkαs. Using condition (8), there gives ω(xk+1, xk+2 ) ≥ ω +(xk+1αs, xk+2αs) > ω+(xk+1, xk+2 ) This is a clear contradiction. Hence, αs ∈ OETn. ii. Let αs ∈ OETn. Suppose that Fo(αs) is empty, that is, there is no fixed β such that β ∈ Fo(αs), this implies that ω+(β,βαs) > ε, for ε > 0. Let ω+(xkαs,β) = 0, by condi- tion (8), there results ω(xk,β) < ω(xkαs,βαs) ≤ ω(xkαs,β) + ω(β,βαs) = ω(β,βαs) On the other hand, since αt is continuous on X for each t ∈ S . Then, for xk ∈ X and τ > 0, ω(xk,β) > τ implies that ω(xkαt,βαt) > ε 2 . Since ω+(xkαs,β) = 0, then there exists b ∈ S such that ω+(xkαab,β) > max{ ε 2 ,δ} for each a ∈ S . Intuitively, ω+(xkαtαab,βαt) > ε 2 . But then, ω+(βαt,β) ≤ ω +(βαt, xkαtab) + ω +(xkαtab,β) and ω+(βαt, xkαtab) + ω +(xkαtab,β) > ε From the last two inequalities, we obtain ω+(βαt,β) = ε. This is a contradiction. Therefore, ω+(βαs,β) = 0 for each s ∈ S and β ∈ Fo(αs). Remark 5. The convergence theorems (strong convergence and ∆-convergence) of the map αs ∈ ℘ satisfying the nonexpansive semigroup of order-preserving total mapping (7) in waist metric spaces can be routinely proved using the next lemma and their left. Lemma 5. Let X be a closed subset of a convex waist met- ric space (M,ω). Suppose αs is a nonexpansive semigroup of order-preserving total mapping (7) of X into itself with Fo(αs) , ∅. Then lims ω(αs x, y) exists for each y ∈ Fo(αs) and s ∈ S . 4. Practical Examples The following three examples are considered to justify the theorems in the main results. The first two examples are from the same family for n = 2 and n = 3, respectively, while the third example is given for n = 3. Example 3. Let S be a semigroup and X = {1, 2}. Let αs : {1, 2}→ {1, 2} be the mapping αs = (1)(2) in OT2, where s ∈ S , associated to xαs = x2 − 2x + 2 in X ⊂ (M,ω)α. The map αs is a nonexpansive order-preserving total map- ping. Hence, it satisfies all hypotheses of Theorem 2 and 4 and the set Fo(αs) = {1, 2}. 5 Usamot et al. / J. Nig. Soc. Phys. Sci. 4 (2023) 878 6 Example 4. Let αs be a mapping in OT3 define by α : {1, 2, 3} → {1, 2, 3} as α = (1)[3 2 1) which is equivalent to the map xαs = x2−3x 2 + 2. Here, the map αs is also a nonexpansive order-preserving total mapping, that is, αs ∈ ONT3. Thus, it satisfies all hy- potheses of Theorem 2 and 4. The set Fo(αs) = {1}. Example 5. Let αs : {1, 2, 3} → {1, 2, 3} be given by the mapping α = (1)[2 1)(3) in OT3 corresponding to the map xαs = x2 − 3x + 3. In Example 5, all hypotheses of Theorem 6 are satisfied since αs ∈ OETn with fixed point Fo(αs) = {1, 3}. 5. Concluding Remark This study introduced the nonexpansive and expansive semigroup of order-preserving total mappings to prove some fixed point theorems in complete waist metric spaces. The study also considered some examples (see Examples 3, 4 and 5) on semigroup of order-preserving total mappings to validate the hypotheses of Theorems 4, 5 & 6. Results show that the nonexpansive (and expansive) semigroup of order-preserving mappings compares favorably with the semigroup of mappings OCTn (and OC∗Tn). In fact, every mapping αs ∈ ℘ under the action of subsemigroups OCTn and OC∗Tn is also in ONTn and OETn, respectively. However, this study only features some elements of subsemigroups of order-preserving full map- pings in [27]. Therefore, future studies would be to establish some notions to study other elements of subsemigroups in ℘ such as order-preserving partial mappings [1, 28, 15], order de- creasing full mappings [29], symmetric inverse semigroups [5], orientation-preserving mappings [30], fence-preserving map- pings [31] among others. Acknowledgements The authors wish to thank Prof. K. Rauf for his mentor-ship. References [1] G. U. Garba, “Nilpotents in semigroups of partial one-to-one order- preserving mappings”, Semigroup Forum 41 (1991) 1. [2] P. M. Higgins, Techniques of Semigroup Theory, Oxford University Press (1992). [3] P. M. Higgins, “Combinatorial results for semigroups of order-preserving mappings”, Math. Proc. Camb. Phil. Soc. 113 (1993) 281. [4] J. M. Howie, Fundamentals of semigroup theory, Oxford University Press Inc. New York (1995). [5] A. Umar, “Some combinatorial problems in the theory of symmetric in- verse semigroups”, Algebra Discrete Math. 9 (2010) 115. [6] R. D. Holmes, A. T. Lau, “Nonexpansive actions of topological semi- groups and fixed points”, J. Lond. Math. Soc. 5 (1972) 330. [7] H. S. Kim, T. H. Kim, “Asymptotic behavior of semigroups of asymptot- ically nonexpansive type on Banach spaces”, J. Korean Math. Soc. 24 (1987) 169. [8] A. T. Lau, Y. Zhang, “Fixed point properties of semigroups of non- expansive mappings”, J. Funct. Anal. 254 (2008) 2534. [9] W. Phuengrattana, S. Suantai, “Fixed point theorems for a semi- group of generalized asymptotically nonexpansive mappings in CAT(0) spaces”, Fixed Point Theory and Applications 2012 (2012) 230. DOI: 10.1186/1687-1812-2012-230. [10] A. H. Soliman, “A tripled fixed point theorem for semigroups of Lip- schitzian mappings on metric spaces with uniform normal structure”, Fixed point theory and Appl. 346 (2013) 1. [11] W. Takahashi, P.J. Zhang, “Asymptotic behavior of almost-orbits of semi- groups of Lipschitzian mappings”, J. Math. Anal. Appl. 142 (1989) 242. [12] M. T. Kiang, “Fixed point theorems for certain classes of semigroups of mappings”, Transactions of the American Mathematical Society 189 (1974) 63. [13] J. C. Yao, L. C. Zeng, “Fixed point theorem for asymtotically regular semigroups in metric spaces with uniformly normal structure”, J. Non- linear Convex Anal. 8 (2007) 153. [14] G. Ayik, H. Ayik, M. Koc, “Combinatorial results for order-preserving and order-decreasing transformations”, Turk J. Math. 35 (2011) 617. [15] A. Laradji, “Fixed points of order-preserving transformations”, Journal of Algebra and Its Applications, 21 (2022) 2250082. [16] S. Banach, “Surles Operations dans les eusembles abstraits et leur appli- cation aus equations integrales”, Fund. Math. 3 (1922) 133. [17] L. P. Belluce, W. A. Kirk, “Fixed point theorem for families of contraction mappings”, Pac. J. Math. 18 (1966) 213. [18] L. P. Belluce, W. A. Kirk, “Nonexpansive mappings and fixed point in Banach spaces”, III. J. Math. 11 (1967) 474. [19] F. E. Browder, “Nonexpansive of nonlinear operators in a Banach space”, Proc. Natl. Sci. USA. 54 (1965) 1041. [20] T. C. Lim, “A fixed point theorem for families of non-expansive map- pings”, Pac. J. Math. 53 (1974) 487. [21] Y. Ibrahim, “Strong convergence theorems for split common fixed point problem of Bregman generalized asymptotically nonexpansive mappings in Banach spaces”, J. Nig. Soc. Phy. Sci. 1(2) (2019) 35 [22] R. Hopkins, Finite metric spaces and their embedding into Lebesgue spaces, University of Chicago Mathematics REU (2015). [23] M. D. Guay, K. L. Singh, J. H. M. White, “Fixed point theorems for non-expansive mappings in convex metric spaces”, Proc. Conference on Nonlinear Analysis (Ed. S. P. Singh and J. H. Bury) Marcel Dekker 80 (1982) 179. [24] W. A. Takahashi, “A convex in metric space and non-expansive mappings I”, Kodai Math. Sem. Rep. 22 (1970) 142. [25] C. E. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Verlag Series: Springer (2009). [26] F. Bruhat, J. Tits, “Groupes r’eductifs sur un corps local. I. Donn’ees radicielles valu’ees”, Inst. Hautes´E tudes Sci. Publ. Math. 41 (1972) 5. [27] A. Laradji & A. Umar, “Combinatorial results for semigroups of order- preserving full transformations”, Semigroup Forum 72 (2006) 51 Springer-Verlag. [28] A. Laradji, A. Umar, “Combinatorial results for semigroups of order- preserving partial transformations”, Journal of Algebra, 278 (2004), 342. [29] A. Laradji, A. Umar, “On certain finite semigroups of order-decreasing transformations I”, Semigroup Forum 69 (2004) 184. [30] I. Dimitrova, J. Koppitz, “On relative ranks of the semigroup of orientation-preserving transformations on infinite chains”, Asian- European Journal of Mathematics, 14 (2021) 2150146. [31] R. Tanyawong, R. Srithus, R. Chinram, “Regular subsemigroups of the semigroups of transformations preserving a fence”, Asian-European Journal of Mathematics 9 (2016) 1650003. 6