Smashed products of topological left quasigroups. Sergey V. Ludkowski 9 October 2020 Abstract In this article smashed products of topological left quasigroups are scrutinized. Quotients of topological fan quasigroups are investigated. Skew smashed products of fan quasigroups and their structure also are studied. 1 Address: Dep. Appl. Mathematics, MIREA - Russian Technological University, av. Vernadsky 78, Moscow 119454, Russia; e-mail: [email protected] 1 Introduction. Topological quasigroups play an important role in algebra, topology, noncom- mutative geometry, mathematical physics [1, 2, 3, 4]. They appear naturally from other areas of mathematics and its applications such as nonassociative arXiv:2012.06636v1 [math.GR] 11 Dec 2020 algebra, quantum field theory, topological algebra [5, 6, 7, 8, 9]. Quasigroups are nonassociative generalizations of groups [4, 10, 11, 12]. A remarkable fact was proved in the 20-th century that the nontrivial geometry exists if and only if there exists the corresponding quasigroup. 1 key words and phrases: topological quasigroup; T1; nonlocally compact; product; smashed Mathematics Subject Classification 2010: 54H11; 54E15; 54D45; 43A05; 28C10; 20N05; 22A30 1 In the article [13] an existence of a left (or right) invariant nontrivial measure on a topological quasigroup was investigated. It was proved that if under rather general conditions the nontrivial left invariant measure exists on the Borel σ-algebra of the topological unital quasigroup G, then G is locally compact. This article is devoted different types of products of quasigroups. In Section 2 direct and smashed products of topological left quasigroups are scrutinized. Quotients of topological fan quasigroups are investigated. Skew smashed products of fan quasigroups and their structure also are studied. Examples are given. It is proved that in particular smashed products of non- trivial topological groups give topological quasigroups which may be nonlo- cally compact. Basic facts on topological quasigroups are given in Appendix. The obtained results permit to construct ample classes of topological quasi- groups. All main results of this paper are obtained for the first time. 2 Products of topological quasigroups. Definition 1. Let G be a left (or right) quasigroup. We put: Com(G) := {a ∈ G : ∀b ∈ G, ab = ba}; (1) Nl(G) := {a ∈ G : ∀b ∈ G, ∀c ∈ G, (ab)c = a(bc)}; (2) Nm(G) := {a ∈ G : ∀b ∈ G, ∀c ∈ G, (ba)c = b(ac)}; (3) Nr(G) := {a ∈ G : ∀b ∈ G, ∀c ∈ G, (bc)a = b(ca)}; (4) N(G) := Nl(G) ∩ Nm(G) ∩ Nr(G); (5) C(G) := Com(G) ∩ N(G). (6) Then N(G) is called a nucleus of G and C(G) is called the center of G. Theorem 1. Let (Gj, τj) be a family of topological left quasigroups, where Gj is the left quasigroup, τj is the topology on Gj, j ∈ J, J is a set. Then their direct product G = Qj∈J Gj relative to the Tychonoff product topology τG is a topological left quasigroup and N(G)= Qj∈J N(Gj) (7) and C(G)= Qj∈J C(Gj). (8) 2 Moreover, if (Gj, τj) is T1 for each j ∈ J, then (G, τ) is T1, N(G) and C(G) are closed in G. Proof. The direct product of left quasigroups satisfies condition (67), hence G is the left quasigroup. Multiplication and Divl are jointly continuous relative to the Tychonoff product topology, consequently, G is the topological left quasigroup. Note that each element a ∈ G can be written as a = {aj : ∀j ∈ J, aj ∈ Gj}. From Formulas (1)-(4) in Definition 1 we deduce that Com(G) := {a ∈ G : ∀b ∈ G, ab = ba} = Qj∈J Com(Gj), (9) since each a and b in G have the forms a = {aj : ∀j ∈ J, aj ∈ Gj} and b = {bj : ∀j ∈ J, bj ∈ Gj}, and since ab = ba if and only if ajbj = bjaj for all j ∈ J; Nl(G) := {a ∈ G : ∀b ∈ G, ∀c ∈ G, (ab)c = a(bc)} = Qj∈J Nl(Gj), (10) since (ab)c = a(bc) if and only if (ajbj)cj = aj(bjcj) for all j ∈ J, where c ∈ G has the form c = {cj : ∀j ∈ J, cj ∈ Gj}; and analogously Nm(G)= Qj∈J Nm(Gj) (11) and Nr(G)= Qj∈J Nr(Gj). (12) Thus formulas (7) and (8) follow from (9)-(12), (5) and (6). If (Gj, τj) is T1 for each j ∈ J, then (G, τ) is T1, since according to Theorem 2.3.11 in [14] a product of T1 spaces is a T1 space. From the joint continuity of multiplication it follows that Com(Gj), Nl(Gj), Nm(Gj), Nr(Gj) are closed in (Gj, τj), hence N(Gj) and C(Gj) are closed in (Gj, τj) for each j ∈ J, consequently, N(G) and C(G) are closed in (G, τ). Theorem 2. Let (A, τA) and (B, τB) be left topological quasigroups, let also ξi : A × B × A → B and A × B ∋ (a, b) 7→ φj(a)b ∈ B be (jointly) continuous univalent maps for each i ∈ {1, 2} and j ∈ {1, 2, 3}, let µ : (A × B)2 → A × B be a map such that (a2) {a1} a1 µ((a1, b1), (a2, b2)) = ((a1, a2), [(ξ1(a1, b1, a2)b1 )ξ2(a1, b1, a2)] b2 ) (13) for each a1, a2 in A; b1, b2 in B, where a1 b2 := φ1(a1)b2, (14) (a2) b1 := φ2(a2)b1, (15) {a1} b2 := φ3(a1)b2, (16) 3 φj : A → A(B), where A(B) denotes the family of all homeomorphisms from B onto B. Then the cartesian product C = A×B supplied with the Tychonoff product topology τC and the map µ is a topological left quasigroup (C, τC). Proof. Multiplications in A and B are univalent, consequently, µ is univalent and provides multiplication in C. Since ξi(a1, b1, a2), φj(a)b are (jointly) continuous for each i ∈{1, 2} and j ∈{1, 2, 3}, (A, τA) and (B, τB) are the left topological quasigroups, then µ is (jointly) continuous. Then we consider the equation: µ((a, b), (x, y))=(c,d), (17) where a ∈ A, c ∈ A, b ∈ B, d ∈ B are arbitrary fixed, x ∈ A, y ∈ B are elements to be expressed through a,b,c,d. The equation (17) is equivalent to the system: ax = c (18) (x) {a} a and [(ξ1(a, b, x)b )ξ2(a, b, x)] y = d. (19) From (18) it follows that x = a \ c, (20) since A is the left quasigroup. Then (19) and (20) imply that (a\c) {a} a [(ξ1(a, b, a \ c)b )ξ2(a, b, a \ c)] y = d; (21) a (a\c) {a} consequently, y = z with z = [(ξ1(a, b, a \ c)b )ξ2(a, b, a \ c)] \ d (22) −1 and hence y = [φ1(a)] z, (23) −1 since φ1(a) ∈ A(B) and [φ1(a)] ∈ A(B). Thus the equation (17) has a unique solution (x, y) given by (20), (23). Denoting it (x, y)=(a, b) \ (c,d) (24) we get that C is the left quasigroup. From the (joint) continuity of µ, ξi, φj for each i ∈{1, 2} and j ∈{1, 2, 3} and Formulas (20), (22)-(24) it follows, that the map Divl is (jointly) continuous, where Divl((a, b), (c,d))=(a, b)\(c,d). Thus (C, τC ) is the topological left quasigroup. Definition 2. The topological left quasigroup provided by Theorem 2 will also be denoted by C = A℘ξ1,ξ2,φ1,φ2,φ3 B and called a smashed product of topological left quasigroups (with smashing factors ξ1,ξ2,φ1,φ2,φ3). Remark 1. Suppose that the conditions of Theorem 2 are satisfied. It 4 is interesting to study another equation: µ((x, y), (a, b))=(c,d), (25) where a, c in A, b, d in B are arbitrarily fixed, x in A and y in B to be found as a solution if it exists. It is equivalent to the system: xa = c (26) (a) {x} x and [(ξ1(x, y, a)y )ξ2(x, y, a)] b = d. (27) Suppose in addition that A is a quasigroup. Then there exists a unique x = c/a in A and (27) takes the form: (a) {c/a} c/a [(ξ1(c/a,y,a)y )ξ2(c/a,y,a)] b = d. (28) Even if B is a quasigroup, apparently there may be either no any or many different solutions y of (28) such that (a) ξ1(c/a,y,a)y = f/ξ2(c/a,y,a), (29) −1 c/a where f = [φ3(c/a)] (d/b ). Indeed, there are A and B such that A(B) − Autc(B) =6 ∅ and C(A × B × A, B) − Homc(A × B × A, B) =6 ∅, where Autc(B) denotes the family of all continuous automorphisms of B, Homc(G, B) denotes the family of all continuous homomorphisms from G into B, C(G, B) denotes the family of all continuous maps from G into B, where G is a topological left quasigroups (or a quasigroup). Thus C = A℘ξ1,ξ2,φ1,φ2,φ3 B is the left topological quasigroup, but generally it need not be a right quasigroup. Example 1. Let A and B be any topological quasigroups such that card(A(B) − Autc(B)) ≥ card(Autc(B)) ≥ c (30) and card(C(A × B × A, B) − Homc(A × B × A, B)) ≥ card(Homc(A × B × A, B)) ≥ c, (31) where c = card(R). Therefore, there exist ξi and φj (see Theorem 2 and Remark 1) such that ξi ∈ C(A × B × A, B) − Homc(A × B × A, B) and φj(A) is not contained in Autc(B) for each i ∈ {1, 2} and j ∈ {1, 2, 3}.