It  is  known  that  the  only   nite  topological  spaces  that  are H-spaces are the discrete spaces.  For a  nite poset which is weakly equivalent to an H-space, a generalized multiplication may be found after suitable sub-division.  In this paper we construct minimal models of the
k-fold generalised multiplications  of  circles  in  the  category  of  relational  structures,  including poset models.  In particular, we obtain higher dimensional analogues of a cer-tain  ternary  multiplication  of  crown

Witbooi, Peter J.

Revista de la Unión Matemática Argentina

University of the Western Cape Research Repository

REVISTA DE LAUNIO´N MATEMA´TICA ARGENTINAVol. 54, No. 2, 2013, Pages 29–36Published online: December 14, 2013MULTIPLICATION OF CROWNSP. J. WITBOOIAbstract. It is known that the only finite topological spaces that are H-spaces are the discrete spaces. For a finite poset which is weakly equivalentto an H-space, a generalized multiplication may be found after suitable sub-division. In this paper we construct minimal models of the k-fold generalisedmultiplications of circles in the category of relational structures, includingposet models. In particular, we obtain higher dimensional analogues of a cer-tain ternary multiplication of crowns [Hardie and Witbooi, Topology Appl.154 (2007), no. 10, 2073–2080].1. IntroductionA categorical equivalence between the category of finite posets and the categoryof finite T0 spaces is discussed in the paper [8] of McCord. Results from [8] implythat for a suitable topological space, in particular for a compact polyhedron, thereexists a finite poset model in the sense of weak homotopy equivalence. For therelevant basic notions of algebraic topology we refer the reader to the textbook [7]of J. P. May. Thus for instance, crowns are poset models of the circle. A posetmodel of (a generalized version of) circle multiplication appears in [3] and is used toproduce a finite model of the Hopf map S3 → S2. A ternary multiplication with therelevant associativity property appears in [4]. In view of the result of Stong [9] onhomotopical triviality of a finite topological space that admits a multiplication, itturns out that some subdivision on the domain side is necessary. Thus for instancein [3] the multiplication is a poset map from the product of two 8-point crowns toa 4-point crown.For a given topological space, we may also want to find a finite model of leastcardinality, i.e., a minimal poset model. See the paper [1] of Barmak and Minianin this regard. Modeling of topological spaces is possible in the bigger category Rof relational structures, defined in the sequel. Such models have been described in[5] and in [2] for instance.2010 Mathematics Subject Classification. 06A06; 18B10.Key words and phrases. poset; relational structure; locally finite space; barycentric subdivi-sion; finite model.The author acknowledges a research grant from the South African National ResearchFoundation.2930 P. J. WITBOOIIn this article we shall consistently use N to denote the set of all non-negativeintegers, i.e.,N = N ∪ {0} .We show how a certain sequence of functions µk : Nk → N give rise to finiteposet models of k-fold circle multiplications. In Section 2 we define the functionsµk and observe how they induce the functions mk : Zk4k → Z4. In Section 4 weprove that each mk turns out to be a poset map from a product of (4k)-pointcrowns to a 4-point crown. In this way we obtain poset models of k-fold circlemultiplication. Also in Section 4, we obtain some further relational models ofthe said circle multiplication. The basic structures and models are presented inSection 3.2. Defining the functionNotation 2.1. Recall, N = N ∪ {0}. For k ∈ N we define the following:(a) For each x ∈ Nk, we writex∗ = min {x1, x2, x3, . . . , xk} and x∗ = max {x1, x2, x3, . . . , xk} .(b) We use the notation, for x ∈ Rk, that ‖x‖ denotes the Euclidean norm, and‖x‖0 = max {|x1| , |x2| , . . . , |xk|}.(c) Let M(k) ={x ∈ Nk : x1 ≥ x2 ≥ x3 ≥ · · · ≥ xk}.(d) For x ∈ Nk and d ∈ N we can form a point x× d ∈ Nk+1 in an obvious way.(e) Note that for every x ∈ Nk, it is possible to permute the coordinates of x soas to obtain an element x¯ ∈M(k).Now we define a sequence of functions µk : Nk → N inductively as follows:Item 2.2. For x ∈ N we let µ1(x) = min{x, 4}. Given any x ∈ Nk, let y = x¯. Wedefine λ1, and then inductively the λi for each i ∈ {2, 3, 4, . . . , k}, and finally µk asfollows:λ1(x) = min{4, y1 − y2},λi(x) = min{4i, λi−1(x) + yi − yi+1},µk(x) = λk(x) = min{4k, λk−1(x) + yk}.Immediately we note that if x ∈ Nk and every coordinate of x is even, thenµk(x) is even. Note also that if x ∈ Nk and y ∈ Nk+1 is obtained by inserting 0as an additional coordinate into x (any position), then µk+1(y) = µk(x). Figure 1is a diagrammatical representation of µ2. We form it by starting with a gridrepresenting a subset of N2 and then we replace each point x of N2 by the numberµ2(x). For better clarity we suppress some occurrences of 5 and 7.The next result follows readily, and we omit the proof.Proposition 2.3. Let k > 1, consider any x ∈ Nk and let y = x¯ ∈M(k). Supposethat for each i ∈ {2, 3, 4, . . . , k} we have y1−yi ≤ 4(i−1). Then λk−1(x) = y1−yk.Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013)MULTIPLICATION OF CROWNS 314 6 8 8 8 8 8 84 6 8 8 8 8 8 84 6 7 8 84 6 6 6 6 6 8 84 5 6 8 84 4 4 4 4 6 8 83 3 3 3 4 62 2 2 3 4 6 6 6 61 1 2 3 40 1 2 3 4 4 4 4 4 4Figure 1.Proposition 2.4. Consider a pair of points x, x′ ∈ Nk. Suppose that for somej ∈ {1, 2, . . . , k} we have xj = x′j + 1 > 4k and xi = x′i ≤ x′j for each i 6= j. Thenµk(x) = µk(x′).Proof. The case k = 1 is clearly true. We assume now that k > 1. Considersuch x, x′ ∈ Nk and let y = x¯ and let y′ = x¯′. First note that if y1 − y2 > 4,then y′1 − y′2 ≥ 4. Therefore, λ1(x) = 4 = λ1(x′) and then it easily follows thatµ(x) = µ(x′). Thus henceforth we assume that y1−y2 ≤ 4. We consider two cases.Case 1 : Suppose that for every i ∈ {1, 2, 3, . . . , k − 1} we have that y1−yi ≤ 4(i−1).Then by Proposition 2.3,λk−1(x) = y1 − yk and λk−1(x′) = y′1 − yk.Thereforeµk(x′) = min {4k, λk−1(x′) + yk} = min {4k, yk} = 4k.Similarly, µk(x) = 4k.Case 2 : Suppose that for some t ∈ {1, 2, 3, . . . , k − 1} we have that y1 − yt >4(t− 1), and choose i to be the smallest among such t. Then λi(x) = 4(i− 1) andλi(x′) = 4(i− 1). Thus it turns out that µk(x) = µk(x′). We set the notation for the pivotal result, Theorem 2.5 below. Let Fk = {x ∈Nk : xi ≤ 4k, for each i = 1, 2, . . . , k}. From Fk we can form a new object Gkby identifying certain pairs of points. These pairs are all those of the form {x, y}such that for some index j we have xj = 0 while yj = 4k, and whenever i 6= jthen xi = yi. The set Gk can be regarded as Zk4k. Here Zn denotes, of course, theintegers modulo n.Theorem 2.5. The function µk induces a (well-defined) functionmk : Zk4k → Z4.Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013)32 P. J. WITBOOIProof. We need to prove that whenever a pair of points x and y are identifiedtowards formation of Gk, thenµk(x) ≡ µk(y) mod 4.Since µ1(0) = 0 and µ1(4) = 4, it follows that the case k = 1 of the propositionis true. For higher dimensions it suffices to show that the statement below is true:If x ∈ Nk and l ∈ N with l ≥ 4(k + 1) ≥ x∗, thenµk+1 (x× l) = µk+1 (x× 0) + 4.Now consider any x ∈ Nk and suppose that l ≥ 4(k + 1) ≥ x∗. By repeatedapplication of Proposition 2.4 we observe that for any x ∈ Nk,µk+1(x× (4k + 4)) = µk+1(x× (4k + 5)) = µk+1(x× (4k + 6)) = . . . .Thus it suffices to assume that l = 4k + 8, in which case λ1(x × l) = 4. Then forevery i ∈ {1, 2, 3, . . . , k} we have thatλi+1(x× l) = 4 + λi(x× 0),and in particular then,µk+1 (x× l) = λk+1(x× l) = 4 + λk(x× 0) = 4 + µk+1(x× 0).3. Relational structuresDefinition 3.1. (See Larose and Tardif [6]). A reflexive binary relational structureX = (X, θ) is a set X equipped with a reflexive binary relationθ ⊆ X ×X.The category of which the objects are the binary reflexive relational structures andthe morphisms are the functions f : (X, θX) → (Y, θY ) satisfying the condition:(x1, x2) ∈ θX ⇒ (fx1, fx2) ∈ θY , will be denoted by R. For short we shall refer toR as the category of relational structures.The category Poset of partially ordered sets (or simply posets) is a full sub-category of R, and provides an important connection between R and the categoryTop of topological spaces and continuous functions.For any finite poset X and any x ∈ X, let Ux = {y ∈ X : (y, x) ∈ θX}. LetT (X) be the topology generated by the subbase U = {Ux : x ∈ X}. If V is anyfinite topological space that satisfies the T0 separation axiom, then we can forman R-object R(V ). We define a relation θV on V as follows: (x, y) ∈ θV if and onlyif y belongs to the closure of the set {x}. In this way we obtain functors T andR which constitute an equivalence between the categories of finite posets and thefinite T0-spaces. Therefore we can refer to a finite poset as a topological space.Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013)MULTIPLICATION OF CROWNS 33Definition 3.2. (See [6]). The product of two relational structures (X, θX) and(Y, θY ) is defined to be the relational structure (X × Y, θX×Y ), where θX×Y is therelation on X × Y described as follows:For x, x1 ∈ X and y, y1 ∈ Y ,((x, y), (x1, y1)) ∈ θX×Y if and only if (x, x1) ∈ θX and (y, y1) ∈ θY .Example 3.3. We consider three different relational structures on Z:(a) Let X0 = (Z, θ0) have the poset relation, or partial order, given by(k, l) ∈ θ0 ⇔ l = k or |k − l| = 1 and k is even.(b) Let X1 = (Z, θ1) be the (antisymmetric) relational structure on the set Z, forwhich we consider (n,m) ∈ θ1 if and only if m− n ∈ {0, 1}.(c) Let X2 = (Z, θ2) be the (symmetric) relational structure on the set Z, for whichwe consider (n,m) ∈ θ2 if and only if |m− n| ≤ 1.Notation 3.4. (a) For any of the relational structures θt defined in Example 3.3,there is an induced structure on subsets and products of subsets. Thus we haveinduced relational structures on Nk. These induced relational structures will bedenoted by the same symbol.(b) In the rest of the paper, for x, y ∈ Nk the statement (x, y) ∈ (Nk, θ0) willbe denoted by x → y. Likewise the relations of θ1 and θ2 will be denoted by ⇀and  respectively. By ≤ we shall denote the usual order on R.(c) For k ∈ N and t = 0, 1, 2, let Xt(4k) denote the sub-relational structuredetermined by the subset {0, 1, 2, . . . , 4k} of Xt and let Ct(4k) be the set obtainedfrom Xt(4k) by identifying the points 0 and 4k, together with the structure θt.The poset C0(4k) is called the 4k-point crown.We record the following observations for easy reference.Remark 3.5. Consider any x, y ∈ Nk. Then,x ⇀ y if and only if for each i we have 0 ≤ yi − xi ≤ 1,x y if and only if for each i we have |yi − xi| ≤ 1,x→ y if and only if (‖x− y‖0 ≤ 1 and xi = yi whenever xi is odd).Definition 3.6. (See [6]). Let us fix an R-morphism g : X → Y . A finite subset{x1, x2, . . . , xk} of X is said to be a chain in X if for every i, j ∈ {1, 2, . . . , k} withi < j, we have (xi, xj) ∈ θX . Let X ′ be the collection of all finite chains in X.Then X ′ is a poset (under subset inclusion), called the barycentric subdivision ofX. For a chain C in X, the subset g(C) is a chain in Y , and so there is a posetmorphism g′ : X ′ → Y ′ making barycentric subdivision to be a functor from R toPoset.Definition 3.7. Recall that a morphism g : X → Y in Top is said to be a weakequivalence or more precisely a weak homotopy equivalence if the induced morphismg∗ : pi0(X)→ pi0(Y ) is a bijection and g∗ : pii(X,x)→ pii(Y, g(x)) is an isomorphismof groups for each i ∈ N and each x ∈ X.Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013)34 P. J. WITBOOITwo morphisms g1 : X1 → Y1 and g2 : X2 → Y2 in Top are said to be weaklyhomotopy equivalent if there is a homotopy commutative diagram such as givenbelow, in which the horizontal arrows are weak homotopy equivalences.X1g1X0oo //g0X2g2Y1 Y0oo // Y2Definition 3.8. An R-morphism g : X → Y is an R-model for the Top-morphismf : A→ B if the barycentric subdivision g′ : X ′ → Y ′ (which can be regarded as aTop-morphism) is weakly homotopy equivalent to f .4. The main theoremIn this section we use the notation u to denote the element of Nk for whichevery coordinate is 1. Also, when unambiguous, we write µk as µ.Proposition 4.1. Suppose that x, y ∈ Nk and for some j we have yj = xj + 1while yi = xi for i 6= j. Then µ(x) ≤ µ(y) ≤ µ(x) + 1.Proof. Without loss of generality we can assume that k ≥ 2 and x, y ∈ M(k). Itis just interesting to note that λj−1(y) ≤ λj−1(x) ≤ λj−1(y) + 1. Nevertheless,if j = k it follows immediately that µk(x) ≤ µk(y) ≤ µk(x) + 1. If j ≤ k,then λj(x) ≤ λj(y) ≤ λj(x) + 1 and from this eventually it turns out again thatµk(x) ≤ µk(y) ≤ µk(x) + 1. Theorem 4.2. Consider any t = 0, 1, 2 and k ∈ N. Given the relational structureθt on N , the function µk is an R-morphism and induces an R-morphism m :Ct(4k)k → Ct(4).Proof. We first prove the case t = 1. It suffices to show that if x, y ∈ Nk andx ⇀ y, thenµ(x) ≤ µ(y) ≤ µ(x) + 1.Clearly, µ(x) ≤ µ(x+u) ≤ µ(x)+1. We can find a sequence as below, with y = x(r)for some r,x = x(1) ⇀ x(2) ⇀ x(3) ⇀ · · ·⇀ x(r) = x+ u,such that in view of Proposition 4.1 we haveµ(x(r)) ≤ µ(x(r+1)) ≤ µ(x(r)) + 1.Then we have µ(x) ≤ µ(y) ≤ µ(x+ u) ≤ µ(x) + 1. This settles the (t = 1)-case.Now we prove the (t = 0)-case. Consider any x, y ∈ Nk with x → y. From the(t = 1)-case it follows that |µ(y) − µ(x)| ≤ 1. Now we need to prove that it isimpossible to have:µ(x) being odd while µ(y) is even. (1)Let y◦ be the point withy◦i = xi if yi > xi, and y◦i = yi otherwise.Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013)MULTIPLICATION OF CROWNS 35There exist xev, yev ∈ Nk such that xev and yev have no odd coordinates and withthe following relations:xev ⇀ x ⇀ xev + u, and y ⇀ y◦ ⇀ yev.Furthermore, µ(xev) and µ(yev) are even integers. Now using the latter facttogether with the aforegoing relations, it can be shown that the situation (1) abovecan never arise. This completes the proof of the (t = 0)-case.Finally, the proof of the (t = 2)-case leans on the (t = 1)-case, with argumentssimilar to (but simpler than) the proof of the (t = 0)-case, and we omit the detail.The restriction of the map m to any of the axes of Ckt (4k), for any of therelational structures θt, yields a model of a degree 1 selfmap of the circle, as is thecase for k-fold circle multiplication. Also the diagonal D = {(z, z, . . . , z) ∈ Sk}is homeomorphic to a circle and the k-fold circle multiplication, when restrictedto D, yields a circle map of degree k. In the discrete case, for each t = 0, 1, 2,the diagonal Dt of Ckt (4k) is isomorphic in R to the crown Ct(4k). Furthermore,the morphism m|Dt is a model of a circle map of degree k. Also note that anysub-relational structure of Ct(4) having fewer than 4 points, will fail to be a modelof the circle.Now let us focus on m in the poset case (and note that the same argument worksfor the other two relational structures covered in Theorem 4.2). Given that thecodomain of the multiplication is a crown with 4 points, and requiring the domainto be a k-fold cartesian power Ckt (l) with the induced map from the diagonal ofCk0 (l) to C0(4) to be a degree k circle map, we can ask how small we can choose l.Simple arithmetic shows that l cannot be chosen any smaller than 4k. Our map mdoes in fact realize a multiplication in this minimal case l = 4k.Acknowledgement. The author wishes to thank the anonymous referee forcomments that led to this improved version of the paper.References[1] J.A. Barmak and E.G. Minian, Minimal finite models, J. Homotopy Relat. Struct. 2 (2007),no. 1, 127–140.[2] A.V. Evako, Topological properties of closed digital spaces: one method of constructing digitalmodels of closed continuous surfaces by using covers, Computer Vision and Image Understand-ing 102 (2006), 134–144.[3] K.A. Hardie, J.J.C. Vermeulen and P.J. Witbooi, A nontrivial pairing of finite T0-spaces,Topology Appl. 125 (2002), no. 3, 533–542.[4] K.A. Hardie and P.J. Witbooi, Crown multiplications and a higher order Hopf construction,Topology Appl. 154 (2007), no. 10, 2073–2080.[5] K.A. Hardie and P.J. Witbooi, Finite relational structure models of topological spaces andmaps, Theoretical Computer Science 405 (2008), 24–34.[6] B. Larose and C. Tardif, A discrete homotopy theory for binary reflexive structures, Adv.Math., 189 (2004), no. 2, 268–300.Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013)36 P. J. WITBOOI[7] J.P. May, A Concise Course in Algebraic Topology, University of Chicago Press, Chicago,1999.[8] M.C. McCord, Singular homology groups and homotopy groups of finite topological spaces,Duke Math. J. 33 (1966), 465–474.[9] R.E. Stong, Finite topological spaces, Trans. Amer. Math. Soc. 123 (1966), 325–340.P. J. WitbooiUniversity of the Western Cape, Bellville 7535, South Africapwitbooi@uwc.ac.zaReceived: November 7, 2011Accepted: March 6, 2013Rev. Un. Mat. Argentina, Vol. 54, No. 2 (2013)

Multiplication of Crowns

It is known that the only  nite topological spaces that are H-
spaces are the discrete spaces. For a  nite poset which is weakly equivalent
to an H-space, a generalized multiplication may be found after suitable sub-
division. In this paper we construct minimal models of the k-fold generalised
multiplications of circles in the category of relational structures, including
poset models. In particular, we obtain higher dimensional analogues of a cer-
tain ternary multiplication of crowns [Hardie and Witbooi, Topology Appl.
154 (2007), no. 10, 2073{2080].Web of Scienc

Witbooi, P. J.

http://repository.uwc.ac.za/xmlui/bitstream/10566/2381/1/Witbooi_Multiplication_2013.pdf

Multiplication of Crowns

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