� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Article in press Available online November 23, 2022 Topologies, posets and finite quandles M. Elhamdadi 1, T. Gona 2, H. Lahrani 1 1 Department of Mathematics and Statistics, University of South Florida Tampa, FL 33620, U.S.A. 2 Department of Mathematics, University of California Berkeley, CA 94720, U.S.A. emohamed@math.usf.edu , gonatushar@berkeley.edu , lahrani@usf.edu Received September 27, 2022 Presented by M. Mbekhta Accepted October 31, 2022 Abstract: An Alexandroff space is a topological space in which every intersection of open sets is open. There is one to one correspondence between Alexandroff T0-spaces and partially ordered sets (posets). We investigate Alexandroff T0-topologies on finite quandles. We prove that there is a non-trivial topology on a finite quandle making right multiplications continuous functions if and only if the quandle has more than one orbit. Furthermore, we show that right continuous posets on quandles with n orbits are n-partite. We also find, for the even dihedral quandles, the number of all possible topologies making the right multiplications continuous. Some explicit computations for quandles of cardinality up to five are given. Key words: quandles, topology, poset. MSC (2020): 54E99, 57K12. 1. Introduction Quandles are algebraic structures modeled on the three Reidemeister moves in classical knot theory. They have been used extensively to construct invariants of knots and links, see for example [6, 8, 10]. A Topological quandle is a quandle with a topology such that the quandle binary operation is compat- ible with the topology. Precisely, the binary operation is continuous and the right multiplications are homeomorphisms. Topological quandles were intro- duced in [11] where it was shown that the set of homomorphisms (called also the set of colorings) from the fundamental quandle of the knot to a topological quandle is an invariant of the knot. Equipped with the compact-open topol- ogy, the set of colorings is a topological space. In [5] a foundational account about topological quandles was given. More precisely, the notions of ideals, kernels, units, and inner automorphism group in the context of topological quandle were introduced. Furthermore, modules and quandle group bundles ISSN: 0213-8743 (print), 2605-5686 (online) c© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0) mailto:emohamed@math.usf.edu mailto:gonatushar@berkeley.edu mailto:lahrani@usf.edu https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 2 m. elhamdadi, t. gona, h. lahrani over topological quandles were introduced with the purpose of studying cen- tral extensions of topological quandles. Continuous cohomology of topolog- ical quandles was introduced in [4] and compared to the algebraic theories. Extensions of topological quandles were studied with respect to continuous 2- cocycles, and used to show differences in second cohomology groups for some specific topological quandles. Nontriviality of continuous cohomology groups for some examples of topological quandles was shown. In [2] the problem of classification of topological Alexander quandle structures, up to isomorphism, on the real line and on the unit circle was investigated. In [7] the author inves- tigated quandle objects internal to groups and topological spaces, extending the well-known classification of quandles internal to abelian groups [13]. In [14] quandle modules over quandles endowed with geometric structures were studied. The author also gave an infinitesimal description of certain modules in the case when the quandle is a regular s-manifold (smooth quandle with certain properties). Since any finite T1-space is discrete, the category of finite T0-spaces was considered in [12], where the point set topological properties of finite spaces were investigated. The homeomorphism classification of finite spaces was investigated and some representations of these spaces as certain classes of matrices was obtained. This article arose from a desire to better understand the analogy of the work given in [12] in the context of finite topological quandles. It turned out that: there is no T0-topology on any finite connected (meaning one orbit un- der the action of the Inner group) quandle X that makes X into a topological quandle (Theorem 4.4). Thus we were lead to consider topologies on finite quandles with more than one orbit. It is well known [1] that the category of Alexandroff T0-spaces is equivalent to the category of partially ordered sets (posets). In our context, we prove that for a finite quandle X with more than one orbit, there exists a unique non trivial topology which makes right mul- tiplications of X continuous maps (Proposition 4.6). Furthermore, we prove that if X be a finite quandle with two orbits X1 and X2 then any continuous poset on X is biparatite with vertex set X1 and X2 (Proposition 4.7). This article is organized as follows. In Section 2 we review the basics of topological quandles. Section 3 reviews some basics of posets, graphs and some hierarchy of separation axioms. In Section 4 the main results of the article are given. Section 5 gives some explicit computations based on some computer software (Maple and Python) of quandles up to order five. topologies, posets and finite quandles 3 2. Review of Quandles and Topological Quandles A quandle is a set X with a binary operation ∗ satisfying the following three axioms: (1) For all x in X, x∗x = x, (2) For all y,z ∈ X, there exists a unique x such that x∗y = z, (3) For all x,y,z ∈ X, (x∗y) ∗z = (x∗z) ∗ (y ∗z). These three conditions come from the axiomatization of the three Reidemeister moves on knot diagrams. The typical examples of quandles are: (i) Any Group G with conjugation x∗y = y−1xy, is a quandle called the conjugation quandle and (ii) Any group G with operation given by x ∗ y = yx−1y, is a quandle called the core quandle. Let X be a quandle. For an element y ∈ X, left multiplication Ly and right multiplication Ry by an element y are the maps from X to X given respectively by Ly(x) := y∗x and Ry(x) = x∗y. A function f : (X,∗) → (X,∗) is a quandle homomorphism if for all x,y ∈ X,f(x∗y) = f(x)∗f(y). If furthermore f is a bijection then it is called an automorphism of the quandle X. We will denote by Aut(X) the automorphism group of X. The subgroup of Aut(X), generated by the automorphisms Rx, is called the inner automorphism group of X and denoted by Inn(X). If the group Inn(X) acts transitively on X, we then say that X is connected quandle meaning it has only one orbit. Since we do not consider topological connectedness in this article, then through the whole article, the word connected quandle will stand for algebraic connectedness. For more on quandles refer to [6, 8, 10, 3]. Topological quandles have been investigated in [2, 5, 11, 4]. Here we review some basics of topological quandles. Definition 2.1. A topological quandle is a quandle X with a topology such that the map X × X 3 (x,y) 7−→ x ∗ y ∈ X is a continuous, the right multiplication Rx : X 3 y 7−→ y ∗x ∈ X is a homeomorphism, for all x ∈ X, and x∗x = x. It is clear that any finite quandle is automatically a topological quandle with respect to the discrete topology. Example 2.2. [2] Let (G, +) be a topological abelian group and let σ be a continuous automorphism of G. The continuous binary operation on G given by x∗y = σ(x) + (Id−σ)(y),∀x,y ∈ G, makes (G,∗) a topological quandle called topological Alexander quandle. In particular, if G = R and σ(x) = tx 4 m. elhamdadi, t. gona, h. lahrani for non-zero t ∈ R, we have the topological Alexander structure on R given by x∗y = tx + (1 − t)y. Example 2.3. The following examples were given in [11, 5]. The unit sphere Sn ⊂ Rn+1 with the binary operation x∗y = 2(x·y)y−x is a topological quandle, where · denotes the inner product of Rn+1. Now consider λ and µ be real numbers, and let x,y ∈ Sn. Then λx∗µy = λ[2µ2(x ·y)y −x]. In particular, the operation ±x∗±y = ±(x∗y) provides a structure of topological quandle on the quotient space that is the projective space RPn. 3. Review of topologies on finite sets, posets and graphs Now we review some basics of directed graphs, posets and T0 and T1 topologies. Definition 3.1. A directed graph G is a pair (V,E) where V is the set of vertices and E is a list of directed line segments called edges between pairs of vertices. An edge from a vertex x to a vertex y will be denoted symbolically by x < y and we will say that x and y are adjacent. The following is an example of a directed graph. Example 3.2. Let G = (V,E) where V = {a,b,c,d} and E = {b < a, c < a, a < d}. d a b c topologies, posets and finite quandles 5 Definition 3.3. An independent set in a graph is a set of pairwise non- adjacent vertices. Definition 3.4. A (directed) graph G = (V,E) is called biparatite if V is the union of two disjoint independent sets V1 and V2. Definition 3.5. A (directed) graph G is called complete biparatite if G is bipartite and for every v1 ∈ V1 and v2 ∈ V2 there is an edges in G that joins v1 and v2. Example 3.6. Let V = V1 ∪ V2 where V1 = {4, 5} and V2 = {1, 2, 3}. Then the directed graph G = (V,E) is complete biparatite graph. Now we recall the definition of partially ordered set. Definition 3.7. A partially ordered set (poset) is a set X with an order denoted ≤ that is reflexive, antisymmetric and transitive. Example 3.8. For any set X, the power set of X ordered by the set inclusion relation ⊆ forms a poset (P(X),⊆) Definition 3.9. Two partially ordered sets P = (X,≤) and Q = (X,≤′) are said to be isomorphic if there exist a bijection f : X → X′ such that x ≤ y if and only if f(x) ≤′ f(y). Definition 3.10. A poset (X,≤) is connected if for all x,y ∈ X, there exists sequence of elements x = x1,x2, . . . ,xn = y such that every two consec- utive elements xi and xi+1 are comparable (meaning xi < xi+1 or xi+1 < xi). Notation: Given an order ≤ on a set X, we will denote x < y whenever x 6= y and x ≤ y. Finite posets (X,≤) can be drawn as directed graphs where the vertex set is X and an arrow goes from x to y whenever x ≤ y. For simplicity, we will not draw loops which correspond to x ≤ x. We will then use the notation (X,<) instead of (X,≤) whenever we want to ignore the reflexivity of the partial order. 6 m. elhamdadi, t. gona, h. lahrani Example 3.11. . Let X = Z8 be the set of integers modulo 8. The map f : X → X given by f(x) = 3x − 2 induces an isomorphism between the following two posets (X,<) and (X,<′). , . Definition 3.12. A chain in a poset (X,<) is a subset C of X such that the restriction of < to C is a total order (i.e. every two elements are comparable). Now we recall some basics about topological spaces called T0 and T1 spaces. Definition 3.13. A topological space X is said to have the property T0 if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other point. Definition 3.14. A topological space X is said to have the property T1 if for every pair of distinct points of X, each point has a neighborhood not containing the other point. Obviously the property T1 implies the property T0. Notice also that this definition is equivalent to saying singletons are closed in X. Thus a T1- topology on a finite set is a discrete topology. Since any finite T1-space is discrete, we will focus on the category of finite T0-spaces. First we need some notations. Let X be a finite topological space. For any x ∈ X, we denote Ux := the smallest open subset of X containing x. It is well known [1] that the category of T0-spaces is isomorphic to the category of posets. We have x ≤ y if and only if Uy ⊆ Ux which is equivalent to Cx ⊂ Cy, where Cv is the complement Ucv of Uv in X. Thus one obtain that Ux = {w ∈ X; x ≤ w} and Cx = {v ∈ X; v < x}. Under this correspondence of categories, the subcategory of finite posets is equivalent to the category of finite T0-spaces. Through the rest of this article we will use the notation of x < y in the poset whenever x 6= y and x ≤ y. topologies, posets and finite quandles 7 4. Topologies on non-connected quandles As we mentioned earlier, since T1-topologies on a finite set are discrete, we will focus in this article on T0-topologies on finite quandles. A map on finite spaces is continuous if and only if it preserves the order. It turned out that on a finite quandle with a T0-topology, left multiplications can not be continuous as can be seen in the following theorem Theorem 4.1. Let X be a finite quandle endowed with a T0-topology. Assume that for all z ∈ X, the map Lz is continuous, then x ≤ y implies Lz(x) = Lz(y). Proof. We prove this theorem by contradiction. Let X be a finite quandle endowed with a T0-topology. Assume that x ≤ y and Lz(x) 6= Lz(y). If x = y, then obviously Lz(x) = Lz(y). Now assume x < y, then for all a ∈ X, the continuity of La implies that a∗x ≤ a∗ y. Assume that there exist a1 ∈ X such that, z1 := a1 ∗ x = La1 (x) < a1 ∗ y = La1 (y). The invertibility of right multiplications in a quandle implies that there exist unique a2 such that a2 ∗ x = a1 ∗ y hence a1 ∗ x < a2 ∗ x which implies a1 6= a2. Now we have a1∗x < a2∗x ≤ a2∗y = z2. We claim that a2∗x < a2∗y. if a2∗y = a2∗x and since a2 ∗x = a1 ∗y we will have a2 ∗y = a2 ∗x = a1 ∗y hence a2 ∗y = a1 ∗y but a1 6= a2, thus contradiction. Now that we have proved a2 ∗ x < a2 ∗ y, then there exists a3 such that a2 ∗y = a3 ∗x we get, a2 ∗x < a3 ∗x repeating the above argument we get, a3 ∗ x < a3 ∗ y. Notice that a1,a2 and a3 are all pairwise disjoint elements of X. Similarly, we construct an infinite chain, a1 ∗x < a2 ∗x < a3 ∗x < · · · , which is impossible since X is a finite quandle. Thus we obtain a contradiction. We have the following Corollary Corollary 4.2. Let X be a finite quandle endowed with a T0-topology. If C is a chain of X as a poset then any left continuous function Lx on X is a constant function on C. Definition 4.3. A quandle with a topology in which right multiplications (respectively left multiplications) are continuous is called right topological quandle (respectively left topological quandle). In other words, right topological quandle means that for all x,y,z ∈ X, x < y ⇒ x∗z < y ∗z. 8 m. elhamdadi, t. gona, h. lahrani and, since left multiplications are not necessarily bijective maps, left topolog- ical quandle means that for all x,y,z ∈ X, x < y ⇒ z ∗x ≤ z ∗y. Theorem 4.4. There is no T0-topology on a finite connected quandle X that makes X into a right topological quandle. Proof. Let x < y. Since X is connected quandle, there exists φ ∈ Inn(X) such that y = φ(x). Since X is finite, φ has a finite order m in the group Inn(X). Since φ is a continuous automorphism then x < φ(x) implies x < φm(x) giving a contradiction. Corollary 4.5. There is no T0-topology on any latin quandle that makes it into a right topological quandle. Thus Theorem 4.4 leads us to consider quandles X that are not connected, that is X = X1 ∪X2 ∪ . . .Xk as orbit decomposition, search for T0-topology on X and investigate the continuity of the binary operation. Proposition 4.6. Let X be a finite quandle with orbit decomposition X = X1 ∪{a}, then there exist unique non trivial T0-topology which makes X right continuous. Proof. Let X = X1 ∪{a} be the orbit decomposition of the quandle X. For any x,y ∈ X1, there exits φ ∈ Inn(X) such that φ(x) = y and φ(a) = a. Declare that x < a, then φ(x) < a. Thus for any z ∈ X1 we have z < a. Uniqueness is obvious. The T0-topology in Proposition 4.6 is precisely given by x < a for all x ∈ X1. Proposition 4.7. Let X be a finite quandle with two orbits X1 and X2. Then any right continuous poset on X is biparatite with vertex set X1 and X2. Proof. We prove this proposition by contradiction. For every x1,y1 ∈ X1 such that x1 < y1. We know that there exist φ ∈ Inn(X) such that φ(x1) = y1. Hence, x1 < φ(x1) implies x1 < φ m(x1) = x1, where m is the order of φ in Inn(X). Thus we have a contradiction. topologies, posets and finite quandles 9 Proposition 4.8. Let X be a finite quandle with two orbits X1 and X2. Then the complete bipartite graph with vertex set X1 and X2 forms a right continuous poset. Proof. Let X be a finite quandle with two orbits X1 and X2. If x ∈ X1 and y ∈ X2 then for every φ ∈ Inn(X) we have φ(x) ∈ X1 and φ(y) ∈ X2. Proposition 4.7 gives that the graph is bipartite and thus x < y. We then obtain φ(x) < φ(y) giving the result. Remark 4.9. By Proposition 4.8 and Theorem 4.1, there is a non-trivial T0-topology making X right continuous if and only if the quandle has more than one orbit. Notice that Proposition 4.8 can be generalized to n-paratite complete graph. The following table gives the list of right continuous posets on some even dihedral quandles. In the table, the notation (a,b) on the right column means a < b. Table 1: Right continuous posets on dihedral quandles Quandle Posets R4 ((0,1),(2,1),(0,3),(2,3)) . ((0, 1), (0, 5), (2, 1), (2, 3), (4, 3), (4, 5)) ; R6 ((0,3), (2, 5), (4, 1)) . (2, 7), (4, 7), (6, 1), (6, 3), (0, 5), (2, 5), (4, 1), (0, 3)) ; R8 (( 0, 1), (6, 7), (4, 5), (0, 7), (2, 1), (2,3), (4, 3), (6, 5)) . ((0, 1), (6, 7), (4, 5), (2, 1), (8, 9), (2, 3), (4, 3), (8, 7), (0, 9), (6, 5)) ; R10 ((4, 7), (6, 9), (2, 9), (8, 1), (8, 5), (0, 7), (6, 3), (2, 5), (4, 1), (0, 3)) ; ((2, 7), (8, 3), (0, 5), (4, 9), (6, 1)) . Notice that in Table 1, the dihedral quandle R4 has only one right con- tinuous poset ((0, 1), (2, 1), (0, 3), (2, 3)) which is complete biparatite. While the dihedral quandle R6 has two continuous posets ((0, 1), (0, 5), (2, 1), (2, 3), (4, 3), (4, 5)) and ((0, 3), (2, 5), (4, 1)) illustrated below. 10 m. elhamdadi, t. gona, h. lahrani 3 0 5 2 1 4 3 2 5 0 1 4 Moreover, in Table 1, for R8 the bijection f given by f(k) = 3k−2 makes the two posets isomorphic. The same bijection gives isomorphism between the first two posets of R10. The following Theorem characterizes non complete biparatite posets on dihedral quandles. Theorem 4.10. Let R2n be a dihedral quandle of even order. Then R2n has s + 1 right continuous posets, where s is number of odd natural numbers less than n and relatively non coprime with n Proof. Let X = R2n be the dihedral quandle with orbits X1 = {0, 2, . . . , 2n − 2} and X2 = {1, 3, . . . , 2n − 1}. For every x ∈ X2, we construct a partial order 1. The two posets <1 and