11 Dec 2020 Smashed Products of Topological Left Quasigroups

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arXiv:2012.06636v1 [math.GR] 11 Dec 2020 ahmtc ujc lsicto 00 41;5E5 44;4 54D45; 54E15; 54H11; 22A30 2010: Classification Subject Mathematics smashed a rvdi h 0t etr httennrva emtyexis quasigroup. corresponding geometry the nontrivial exists the there that if century only 20-th the rema A in proved 12]. 11, was 10, Q [4, groups 9]. of 8, generalizations 7, nonassociative 6, are [5, nonas algebra as topological theory, such field applications quantum its algebra, and mathematics of appea areas They other 4]. from 3, 2, [1, physics mathematical geometry, topology,mutative algebra, in role important an play quasigroups Topological Introduction. 1 Russia; 119454, Moscow 78, Vernadsky av. University, 1 drs:Dp pl ahmtc,MRA-RsinTechnological [email protected] Russian e-mail: - MIREA Mathematics, Appl. Dep. Address: e od n hae:tplgclquasigroup; topological phrases: and words key kwsahdpout ffnqairusadterstructur their and studied. quasigroups are fan of products smashed Skew are quasigroups fan topological of Quotients scrutinized. mse rdcso oooia left topological of products Smashed nti ril mse rdcso oooia etquasig left topological of products smashed article this In 1 egyV Ludkowski V. Sergey quasigroups. coe 2020 October 9 Abstract 1 T 1 olclycmat product; compact; nonlocally ; A5 81;20N05; 28C10; 3A05; investigated. op are roups also e naturally r uasigroups kbefact rkable si and if ts sociative noncom- 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 nontrivial topological groups give topological quasigroups which may be nonlocally compact. Basic facts on topological quasigroups are given in Appendix. The obtained results permit to construct ample classes of topological quasigroups. 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}. Hence ξi

ξ1,ξ2,φ1,φ2,φ3 and φj can be chosen such that C = A℘ B is the topological left quasigroup (nonassociative), but not the right quasigroup. Example 2. We take the special orthogonal group A = SO(n, R) of the Euclidean space Rn, the special linear group B = SL(m, R) of the Euclidean space Rm, where 1 < n ≤ m ∈ N, A and B are supplied with topologies induced by the operator norm topology. Therefore for them Conditions (30)

5 and (31) are satisﬁed. Hence there exist topological left quasigroups (nonassociative) C = A℘ξ1,ξ2,φ1,φ2,φ3 B which are not right quasigroups. The topological left quasigroup C is locally compact and locally connected, its small (m2−1)n(n−1) inductive dimension is positive 2 = ind(C).

Example 3. Let l2 be the separable Hilbert space over the complex field C, where C is supplied with the standard multiplicative norm topology. We consider the unitary group A = U(l2) and the general linear group B = GL(l2) of l2, where A and B are considered in the topologies inherited form the operator norm topology. For these groups Conditions (30) and (31) are satisfied. Therefore there exist their smashed products C = A℘ξ1,ξ2,φ1,φ2,φ3 B which are topological left quasigroups (nonassociative), but not right quasigroups. Moreover, C is locally connected and nonlocally compact, ind(C)= ∞. Example 4. Assume that F is an infinite nondiscrete spherically com- plete field supplied with a multiplicative norm satisfying the strong triangle inequality |a + b| ≤ max(|a|, |b|) for each a and b in F (see [15]).

By c0(α, F) = X is denoted a Banach space consisting of all vectors x = (xj : ∀j ∈ α, xj ∈ F) such that for each ǫ> 0 the cardinality card(j ∈

α : |xj| > ǫ) < ℵ0 and with a norm |x| = supj∈α |xj|, where α is a (nonvoid) set. We consider the linear isometry group A = IL(X) and the general linear group B = GL(X) of X supplied with topologies inherited from the operator norm topology, where IL(X) = {g ∈ GL(X): ∀x ∈ X, |g(x)| = |x|}. Evidently, A and B satisfy Conditions (30) and (31), consequently, there exist their smashed products C = A℘ξ1,ξ2,φ1,φ2,φ3 B such that C is a topological left quasigroup and such that C is not a right quasigroup. Therefore C is totally disconnected, since F, X, A and B are totally disconnected. If card(α) < ℵ0 and F is locally compact, then C is locally compact. If either card(α) ≥ ℵ0 or F is not locally compact, then C is not locally compact.

Corollary 1. Let the conditions of Theorem 1 be satisﬁed and let Gj be compact for each j ∈ J0 and locally compact for each j ∈ J − J0, where

J0 ⊂ J. If J −J0 is a ﬁnite set, then G is locally compact. If J −J0 is inﬁnite and Gj is noncompact for each j ∈ J − J0, then G is not locally compact.

6 Proof. By Theorem 1 G is the topological left quasigroup. In view of

Theorem 3.3.13 in [14] G as the topological space is locally compact if J − J0 is finite, G is not locally compact if J − J0 is infinite and Gj is noncompact for each j ∈ J − J0. Corollary 2. Suppose that the conditions of Theorem 2 are satisfied. Then the smashed product C = A℘ξ1,ξ2,φ1,φ2,φ3 B is locally compact if and only if A and B are locally compact. Proof. The assertion of this Corollary is evident by Theorems 3.3.13 in [14] and 2 above. Next topological quasigroups with some mild restrictions are considered. They are particular cases of topological left quasigroups. For them also smashed products are investigated. Definition 3. We call G a fan quasigroup if the unital quasigroup satisfies Conditions (32)-(34): (ab)c = t(a, b, c)a(bc) (32) and (ab)c = a(bc)p(a, b, c) (33) for each a, b and c in G, where

t(a, b, c)= tG(a, b, c) ∈ N(G) and p(a, b, c)= pG(a, b, c) ∈ N(G). (34)

A minimal closed subgroup N0(G) in the topological fan quasigroup G containing t(a, b, c) and p(a, b, c) for each a, b and c in G will be called a fan of G.

Corollary 3. Let the conditions of Theorem 1 be satisﬁed and let Gj be a fan quasigroup for each j ∈ J. Then G is a topological fan quasigroup. Proof. In view of Theorem 1 G is the topological quasigroup. If a, b and c are in G, then

(ab)c = {(ajbj)cj : ∀j ∈ J, aj ∈ Gj, bj ∈ Gj,cj ∈ Gj}

= {tGj (aj, bj,cj)aj(bjcj): ∀j ∈ J, aj ∈ Gj, bj ∈ Gj,cj ∈ Gj}

= tG(a, b, c)a(bc) (35) and

(ab)c = a(bc)pG(a, b, c) (36)

with tG(a, b, c) = {tGj (aj, bj,cj): ∀j ∈ J, aj ∈ Gj, bj ∈ Gj,cj ∈ Gj} (37)

7 and pG(a, b, c) = {pGj (aj, bj,cj): ∀j ∈ J, aj ∈ Gj, bj ∈ Gj,cj ∈ Gj}. (38) Formulas (35)-(38) imply that Conditions (32)-(34) are satisﬁed. On the other hand, Conditions (2)-(5) imply that N(G) is a group. Thus G is the topological fan quasigroup. Deﬁnition 4. Let a subquasigroup H of a quasigroup G satisfy conditions (39) and (40): xH = Hx (39) and (xy)H = x(yH) and (xH)y = x(Hy) and H(xy)=(Hx)y (40) for each x and y in G. Then H is called normal in G.

A family of cosets {bH : b ∈ G} will be denoted by G/cH.

This notation is used below in order to distinguish it from b/a = Divr(a, b). Remark 2. Assume that A and B are two topological fan quasigroups;

N0(A) and N0(B) are fans of A and B respectively. Let also N be a topological group such that

(N0(A) ֒→ N ֒→ N(A) and N0(B) ֒→ N ֒→ N(B) (41 and let N be normal in A and in B (see also Deﬁnitions 1 and 4). Using direct products it is always possible to extend either A or B to get such a case. In particular, either A or B may be a group. On A × B an equivalence relation Ξ is considered such that (vγ,b)Ξ(v,γb) (42) for every v in A, b in B and γ in N. Let φ : A →A(B) be a univalent mapping, (43) where A(B) denotes the family of all homeomorphisms from B onto B. If a ∈ A and b ∈ B, then we will write ba for φ(a)b, where φ(a): B → B. Let also

ηφ : A × A × B → N, κφ : A × B × B → N

and ξφ :(A × B) × (A × B) → N be univalent maps written shortly as η, κ and ξ correspondingly such that (bu)v = bvuη(v,u,b), γu = γ, bγ = b; (44)

η(v, u, (γ1b)γ2)= η(v,u,b); (45) if γ ∈{v,u,b} then η(v,u,b)= e; (cb)u = cubuκ(u,c,b); (46)

8 κ(u, (γ1c)γ2, (γ3b)γ4)= κ(u,c,b) (47) and if γ ∈{u,c,b) then κ(u,c,b)= e;

ξ(((γu)γ1, (γ2c)γ3), ((γ4v)γ5, (γ6b)γ7)) = ξ((u,c), (v, b)) and ξ((e, e), (v, b)) = e and ξ((u,c), (e, e)) = e (48) for every u and v in A, b, c in B, γ, γ1,...,γ7 in N, where e denotes the neutral element in N and in A and B. We put

a1 (a1, b1)(a2, b2)=(a1a2, b2 b1ξ((a1, b1), (a2, b2))) (49) for each a1, a2 in A, b1 and b2 in B. The Cartesian product A × B supplied with such a binary operation (49) will be denoted by A♣φ,η,κ,ξB. Theorem 3. Let the conditions of Remark 2 be fulﬁlled. Then the Carte- sian product A × B = G supplied with the binary operation (49) is a fan quasigroup. Proof. From the conditions of Remark 2 it follows that the binary operation (49) is univalent. The group N is normal in the quasigroups A and B by Conditions (41). Hence for each a ∈ A and β ∈ N we get (aβ)/a ∈ N and a \ (βa) ∈ N, since aN = Na for each a ∈ A. Analogously for B. Thus there are univalent maps

rA,a(β)=(aβ)/a,r ˇA,a(β)= a \ (βa),

rB,b(β)=(bβ)/b,r ˇB,b(β)= b \ (βb),

rA,a : N → N,r ˇA,a : N → N, rB,b : N → N,r ˇB,b : N → N for each a ∈ A and b ∈ B. Evidently

rA,a(ˇrA,a(β)) = β andr ˇA,a(rA,a(β)) = β for each a ∈ A and β ∈ N, and similarly for B.

Let I1 = ((a1, b1)(a2, b2))(a3, b3) and I2 = (a1, b1)((a2, b2)(a3, b3)), where a1, a2, a3 belong to A; b1, b2, b3 belong to B. Then using (44)-(48) we infer that

a1a2 a1 a1 I1 = ((a1a2)a3, b3 (b2 b1)ξ((a1, b1), (a2, b2))ξ((a1a2, b2 b1), (a3, b3))) and

a1a2 a1 I2 =(a1(a2a3), (b3 η(a1, a2, b3)b2 )

a2 a2 κ(a1, b3 , b2)ξ((a2, b2), (a3, b3))b1ξ((a1, b1), (a2a3, b3 b2))). Therefore

a1a2 a1 I1 =(a, bα) with a = a1(a2a3), b =(b3 b2 )b1, a1a2 a1 −1 α =r ˇB,b(pA(a1, a2, a3))[pB(b3 , b2 , b1)]

9 a1 ξ((a1, b1), (a2, b2))ξ((a1a2, b2 b1), (a3, b3)) and

a2 I2 =(a, bβ) with β =r ˇB,b1 (γ)ξ((a1, b1), (a2a3, b3 b2)),

a2 a1 where γ =r ˇB,b2 (η(a1, a2, b3))κ(a1, b3 , b2)ξ((a2, b2), (a3, b3)). Hence

I1 = I2pG with pG = pG((a1, b1), (a2, b2), (a3, b3)) (50)

and I1 = tGI2 with tG = tG((a1, b1), (a2, b2), (a3, b3)); −1 pG = β α and tG = rA,a(rB,b(p)), (51) where G = A♣φ,η,κ,ξB.

Apparently tG((a1, b1), (a2, b2), (a3, b3)) ∈ N and pG((a1, b1), (a2, b2), (a3, b3)) ∈ N for each aj ∈ A, bj ∈ B, j ∈{1, 2, 3}, since α and β belong to the group N.

If γ ∈ N and either (γ, e) or (e, γ) belongs to {(a1, b1), (a2, b2), (a3, b3)}, then from the conditions in Remark 2 and Formulas (50) and (51) it follows that

pG((a1, b1), (a2, b2), (a3, b3)) = e and

tG((a1, b1), (a2, b2), (a3, b3)) = e, consequently, (N, e) ∪ (e, N) ⊂ N(G), hence (N, e)(e, N)=(N, N) ⊂ N(G) by (42). Evidently (69) follows from (48) and (49). Next we consider the following equation (x, y)(a, b)=(e, e), (52) where a ∈ A, b ∈ B are arbitrarily ﬁxed, x ∈ A and y ∈ B to be calculated. From (68) for the fan quasigroups A and B, (48) and (49) it follows that x = e/a, (53) consequently, b(e/a)yξ((e/a, y), (a, b)) = e and hence y = [b(e/a) \ e]/ξ((e/a, b(e/a) \ e), (a, b))]. (54) Thus x ∈ A and y ∈ B given by (53) and (54) provide a unique solution of (52). Similarly from the following equation (a, b)(v, z)=(e, e), (55) where a ∈ A, b ∈ B are arbitrarily ﬁxed, v ∈ A and z ∈ B are to be found, we infer that v = a \ e (56), consequently, zabξ((a, b), (a \ e, z)) = e and hence

10 za = [ξ((a, b), (a \ e, z))]−1/b by Conditions (67), (68) and (43) for the fan quasigroups A and B. Notice that (za)e/a = zη(e/a, a, z), hence by Lemmas 1, 2 and the conditions of Remark 2 z = {[ξ((a, b), (a \ e, (e/b)e/a))]−1/b}e/a/η(e/a, a, (e/b)e/a). (57) Thus Formulas (56) and (57) provide a unique solution of (55). Then it is natural to put (x, y)=(e, e)/(a, b) and (v, z)=(a, b) \ (e, e) and

(a, b) \ (c,d)=((a, b) \ (e, e))(c,d)pG((a, b), (a, b) \ (e, e), (c,d)); (58) −1 (c,d)/(a, b) = [tG((c,d), (e, e)/(a, b), (a, b))] (c,d)((e, e)/(a, b)) (59) φ,η,κ,ξ and eG = (e, e), where G = A♣ B. Therefore Properties (67)-(69) and (32)-(34) are fulﬁlled for G. Deﬁnition 5. We call the fan quasigroup A♣φ,η,κ,ξB provided by Theo- rem 3 a skew smashed product of the fan quasigroups A and B with smashing factors φ, η, κ and ξ.

Theorem 4. If G is a T1 topological fan quasigroup, then its fan N0 =

N0(G) is a normal subgroup. Moreover, if N0 ⊆ N1 ⊆ N(G), N1 is a closed in G subgroup satisfying (39), then its quotient G/cN1 is a T1 ∩T3.5 topological group.

Proof. Let τ be a T1 topology on G relative to which G is a topological quasigroup. Then each point x in G is closed, since G is the T1 topological space (see Section 1.5 in [14]). From the joint continuity of the multiplication and the mappings Divl and Divr it follows that the nucleus N(G) is closed in

G. Therefore the subgroup N0 is the closure of a subgroup N0,0(G) in N(G) generated by elements tG(a, b, c) and pG(a, b, c) for all a, b and c in G (see

Deﬁnitions 1 and 6). According to (2)-(5) one gets that N(G) and hence N0 are subgroups in G satisfying Conditions (40), because N0 ⊆ N(G). Let a and b belong to N(G) and x ∈ G. Then x(x \ (ab)) = ab and x((x \ a)b)=(x(x \ a))b = ab, consequently, x \ (ab)=(x \ a)b for each a and b in N(G), x ∈ G. (60) Similarly it is deduced (ab)/x = a(b/x) for each a and b in N(G), x ∈ G. (61) Therefore from (34) and (80) and (60) it follows that

11 ((x \ a)x)((x \ b)x)=(x \ a)(x((x \ b)x))p(x \ a, x, (x \ b)x) =(x \ (ab))x[p(x, x \ b, x)]−1p(x \ a, x, (x \ b)x), since (x \ a)(bx)=((x \ a)b)x =(x \ (ab))x. Thus (x \ (ab))x = ((x \ a)x)((x \ b)x)[p(x \ a, x, (x \ b)x)]−1p(x, x \ b, x) for each a and b in N(G), x ∈ G. (62) From Identities (73) and (74) it follows that x \ ((u \ v)y)=((ux) \ (vy))p(u, x, (ux) \ (vy))[p(u,u \ v, x)]−1 (63) for each u, v, x and y in G, since x \ ((u \ v)y)= x \ (u \ (vy))[p(u,u \ v,y)]−1. In particular for u = a(bc) and v = (ab)c with any a, b and c in G we infer using (34) that ux = (a(b(cx)))p(b, c, x)p(a, bc, x) and vx = (ab)(cx)p(ab, c, x), hence from (63) and (93) it follows that x\(p(a, b, c)x) = [p(b, c, x)p(a, bc, x)]−1p(a, b, cx)p(u, x, (ux)\(vx)), (64) since x \ (p(a, b, c)x) = [(a(b(cx)))p(b, c, x)p(a, bc, x)] \ [(ab)(cx)p(ab, c, x)] p(u, x, (ux) \ (vx))[p(u,u \ v, x)]−1, because u \ v = p(a, b, c) ∈ N(G) and p(u,u \ v, x)= e. Notice that (67), (68) and (32)-(34) imply u\(tu)= p, where t = t(a, b, c), p = p(a, b, c), u = a(bc) for any a, b and c in G. Let z ∈ G, then there exists x ∈ G such that z = ux, that is x = u \ z. Therefore we deduce that z \ (tz) = [x \ (px)]p(u,u \ (tu), x)[p(u, x, (ux) \ (tux))]−1, (65) since t ∈ N(G), p ∈ N(G), (u \ (tu))x = (u \ (tux))[p(u,u \ (tu), x)]−1 by (74); x \ (u \ (tux)) = [(ux) \ (tux))]p(u, x, (ux) \ (tux)) by (73). Thus from Identities (62), (64) and (65) it follows that a group N0,0 = N0,0(G) generated by {p(a, b, c), t(a, b, c): a ∈ G, b ∈ G, c ∈ G} satisﬁes Condition

(39). From the joint continuity of multiplication and the mappings Divl and

Divr it follows that the closure N0 of N0,0 also satisﬁes (39). Thus N0 is a closed normal subgroup in G.

Since N1 satisﬁes (39) and N1 is the subgroup in G such that N0 ⊆ N1 ⊆

N(G), then N1 is normal in G. Therefore a quotient quasigroup G/cN1 exists consisting of all cosets aN1, where a ∈ G. Then from Conditions (34), (39) and (40) it follows that for each a, b, c in G the identities take place

12 (aN1)(bN1)=(ab)N1 and

((aN1)(bN1))(cN1)=(aN1)((bN1)(cN1)) and eN1 = N1, since pG(a, b, c) ∈ N0 ⊆ N1 and tG(a, b, c) ∈ N0 ⊆ N1 for all a, b and c in G.

In view of Lemmas 1 and 2 (aN1) \ e = e/(aN1), consequently, for each −1 aN1 ∈ G/cN1 a unique inverse (aN1) exists. Thus the quotient G/cN1 of

G by N1 is a group. Since the topology τ on G is T1 and N1 is closed in G by the conditions of this theorem, then the quotient topology τq on G/cN1 is also T1. By virtue of Theorem 8.4 in [16] this implies that τq is a T1 ∩ T3.5 topology on G/cN1. Corollary 4. Suppose that the conditions of Remark 2 are fulﬁlled and A and B are topological T1 fan quasigroups and smashing factors φ, η, κ, ξ are jointly continuous by their variables. Suppose also that A♣φ,η,κ,ξB is supplied with a topology induced from the Tychonoﬀ product topology on A × B. Then

φ,η,κ,ξ A♣ B is a topological T1 fan quasigroup. Remark 3. In particular, it is possible to consider a topological quasigroup G satisfying the condition:

there exists a compact subgroup N0 = N0(G) in N(G) such that

tG(a, b, c) ∈ N0 and pG(a, b, c) ∈ N0 for every a, b and c in G. (66) Corollary 5. If the conditions of Corollary 4 are satisﬁed and quasigroups A and B are locally compact, then A♣φ,η,κ,ξB is locally compact. Moreover, if A and B satisfy Condition (66) and ranges of η, κ, ξ are con-

φ,η,κ,ξ tained in N0(A)N0(B), then A♣ B satisﬁes Condition (66). Proof. Corollaries 4 and 5 follow immediately from Theorems 2.3.11, 3.2.4, 3.3.13 in [14], Lemma 2.6 in [13] and Theorems 3, 4, Lemma 3, since

N0(A)N0(B) ⊆ N ⊆ N(A)∩N(B) and because N0(A)N0(B) is a compact subgroup in A♣φ,η,κ,ξB. Remark 4. From Theorems 1, 3, 4 and Corollaries 4, 5 it follows that taking nontrivial φ, η, κ and ξ and starting even from groups with nontrivial N(Gj) or N(A) and Gj/cN(Gj) or A/cN(A) it is possible to construct new fan quasigroups with nontrivial N0(G) and ranges tG(G,G,G) and pG(G,G,G) of tG and pG may be inﬁnite and nondiscrete. With suitable smashing factors φ, η, κ and ξ and with nontrivial fan quasigroups or groups A and B it is easy to get examples of fan quasigroups in which e/a =6 a \ e

13 for an infinite family of elements a in A♣φ,η,κ,ξB. It can be seen that the smashed product of topological quasigroups is a generalization of a semidirect product of topological groups. Conclusion. The results of this article can be used for further inves- tigations of structure of topological quasigroups, their smashed and skew smashed products, homogeneous spaces associated with quasigroups [17], measures on homogeneous spaces and noncommutative manifolds [11], microbundles [18, 19]. It is worth to mention possible applications in mathematical coding theory, techniques such as information flows analysis and systems with distributed memory [20, 21, 22]. Indeed, codes are frequently based on topological-algebraic binary systems. Naturally, they also can be utilized in harmonic analysis on nonassociative algebras [12], quasigroups, their representation theory [4], geometry, mathematical physics, quantum field theory, gauge theory, PDEs, etc. [5, 8, 23, 24, 25, 26, 27].

3 Appendix. Basics on topological quasi-groups.

For convenience we remind a definition, though a reader familiar with [4, 10, 14] may skip Definition 6. Definition 6. Let G be a set with multiplication (that is a univalent binary operation) G2 ∋ (a, b) 7→ ab ∈ G defined on G such that for each a and b in G there is a unique x ∈ G with ax = b. (67) A set G possessing multiplication and satisfying condition (67) is called a left quasigroup. Symmetrically it is considered a unique y ∈ G exists satisfying ya = b. (68) Then the set G possessing multiplication and satisfying condition (68) is called a right quasigroup.

The maps in (67) and (68) are denoted by x = a \ b = Divl(a, b) and y = b/a = Divr(a, b) correspondingly. If G is a left and right quasigroup, then it is called a quasigroup. Let τ be a topology on the left (or right) quasigroup G such that multiplication G × G ∋ (a, b) 7→ ab ∈ G and the mapping Divl(a, b) (or Divr(a, b) respectively) are jointly continuous relative to τ, then (G, τ) will be called a

14 topological left (or right respectively) quasigroup. If G is a topological left and right quasigroup, then it is called a topological quasigroup. A set G possessing multiplication is called a groupoid.

If there exists a neutral (i.e. unit) element eG = e ∈ G: eg = ge = g for each g ∈ G, (69). then the groupoid G is called unital. If A and B are subsets in G, then A − B means the diﬀerence of them A − B = {a ∈ A : a∈ / B}. Lemma 1. If G is a fan quasigroup, then for each a, b and c in G the following identities are fulﬁlled: b \ e = t(e/b, b, b \ e)(e/b); (70) b \ e =(e/b)p(e/b, b, b \ e); (71) (a \ e)b = t(e/a, a, a \ e)[t(e/a, a, a \ b)]−1(a \ b); (72) (a \ b)=(a \ e)bp(a, a \ e, b); (bc) \ a =(c \ (b \ a))[p(b, c, (bc) \ a)]−1; (73) (a \ b)c =(a \ (bc))[p(a, a \ b, c)]−1; (74) (ab) \ e =(b \ e)(a \ e)[t(a, b, b \ e)]−1t(ab, b \ e, a \ e); (75) b(e/a)=(b/a)p(b/a, a, a \ e)[p(e/a, a, a \ e)]−1; (b/a) = [t(b, e/a, a)]−1b(e/a); (76) a/(bc)= t(a/(bc),b,c)((a/c)/b); (77) c(b/a)= t(c, b/a, a)(cb)/a; (78) e/(ab) = [p(e/b, e/a, ab)]−1p(e/a, a, b)(e/b)(e/a). (79) Proof. Note that N(G) is a subgroup in G due to Conditions (2)-(5). Then Conditions (67)-(69) imply that b(b \ a)= a, b \ (ba)= a; (80) (a/b)b = a, (ab)/b = a (81) for each a and b in any quasigroup G. Using Conditions (32)-(34) and Iden- tities (80) and (81) we deduce that e/b =(e/b)(b(b \ e)) = [t(e/b, b, b \ e)]−1(b \ e) which leads to (70). Let c = a \ b, then from Identities (70) and (80) it follows that (a \ e)b = t(e/a, a, a \ e)(e/a)(ac)

15 = t(e/a, a, a \ e)[t(e/a, a, a \ b)]−1((e/a)a)(a \ b) which taking into account (81) provides (72). On the other hand, b \ e = ((e/b)b)(b \ e)=(e/b)(b(b \ e))p(e/b, b, b \ e) that gives (71). Let now d = b/a, then Identities (71) and (81) imply that b(e/a)=(da)(a \ e)[p(e/a, a, a \ e)]−1 =(b/a)p(b/a, a, a \ e)[p(e/a, a, a \ e)]−1 which demonstrates (76). Next we infer from (32)-(34) and (80) that b(c((bc)\a))=(bc)((bc)\a)[p(b, c, (bc)\a)]−1 = a[p(b, c, (bc)\a)]−1, hence c((bc) \ a)=(b \ a)[p(b, c, (bc) \ a)]−1 that implies (73). Symmetrically it is deduced that (a/(bc))b)c = t(a/(bc),b,c)a, consequently, (a/(bc))b = t(a/(bc),b,c)(a/c). From the latter identity it follows (77). Evidently, formulas a((a \ b)c)=(a(a \ b))c[p(a, a \ b, c)]−1 = bc[p(a, a \ b, c)]−1 and (c(b/a))a = t(c, b/a, a)cb imply (74) and (78) correspondingly. From (34) we infer that (ab)((b \ e)(a \ e)) = [t(ab, b \ e, a \ e)]−1t(a, b, b \ e), since by (80) (a(b(b \ e)))(a \ e)= e. This together with (67) and (68) implies (75). Analogously form (32) we deduce that ((e/b)(e/a))(ab) = [p(e/a, a, b)]−1p(e/b, e/a, ab), since by (81) (e/b)(((e/a)a)b)= e. Finally applying (67) and (68) we get Identity (79).

Lemma 2. Assume that G is a fan quasigroup. Then for every a, a1, a2, a3 in G and z1, z2, z3 in C(G), b ∈ N(G):

t(z1a1, z2a2, z3a3)= t(a1, a2, a3); (82)

p(z1a1, z2a2, z3a3)= p(a1, a2, a3); (83) t(a, a \ e, a)a = ap(a, a \ e, a); (84) t(a, e/a, a)a = ap(a, e/a, a); (85) p(a, a \ e, a)t(e/a, a, a \ e)= e; (86)

16 t(a1, a2, a3b)= t(a1, a2, a3); (87)

p(ba1, a2, a3)= p(a1, a2, a3); (88) −1 t(ba1, a2, a3)= bt(a1, a2, a3)b ; (89) −1 p(a1, a2, a3b)= b p(a1, a2, a3)b. (90)

Proof. Since (a1a2)a3 = t(a1, a2, a3)a1(a2a3) and t(a1, a2, a3) ∈ N(G) for every a1, a2, a3 in G, then

t(a1, a2, a3)=((a1a2)a3)/(a1(a2a3)). (91)

Therefore, for every a1, a2, a3 in G and z1, z2, z3 in C(G) we infer that

t(z1a1, z2a2, z3a3) = (((z1a1)(z2a2))(z3a3))/((z1a1)((z2a2)(z3a3))) =

((z1z2z3)((a1a2)a3))/((z1z2z3)(a1(a2a3))) = ((a1a2)a3)/(a1(a2a3)), (92) since b/(qa)= q−1b/a and b/q = q \ b = bq−1 for each q ∈ C(G), a and b in G, because C(G) is the commutative group satisfying Conditions (1) and (6). Thus t(z1a1, z2a2, z3a3)= t(a1, a2, a3). Symmetrically we get

p(a1, a2, a3)=(a1(a2a3)) \ ((a1a2)a3) and

p(z1a1, z2a2, z3a3)=((z1a1)((z2a2)(z3a3))) \ (((z1a1)(z2a2))(z3a3))

= ((z1z2z3)(a1(a2a3)))\((z1z2z3)((a1a2)a3))=(a1(a2a3))\((a1a2)a3) (93) that provides (83). From Formulas (91) and (70) it follows that t(a, a\e, a)=((a(a\e))a)/(a((a\e)a)) = a/[at(e/a, a, a\e)], consequently, t(a, a \ e, a)at(e/a, a, a \ e)= a. (94) Then from Formulas (93), (80) and Conditions (32)-(34) we deduce that p(a, a \ e, a)=(a((a \ e)a)) \ ((a(a \ e))a)= {[t(a, a \ e, a)]−1a}\ a, which implies (84). Identities (84) and (94) lead to (86). Next using (93) and (34) we deduce that p(a, e/a, a) = [a((e/a)a)] \ [(a(e/a))a]= a \ [t(a, e/a, a)a] that implies (85). From (34) we get that

((a1a2)a3)b =(a1a2)(a3b)=(t(a1, a2, a3b)a1(a2a3))b, from which and (81) and (91) Identity (87) follows, because b ∈ N(G). Then

b((a1a2)a3)=((ba1)a2)a3 = b(a1(a2a3)p(ba1, a2, a3)) and (80) and (93) imply Identity (88). Symmetrically we deduce

b((a1a2)a3)= t(ba1, a2, a3))b(a1(a2a3)) and

17 ((a1a2)a3)b =(a1(a2a3))bp(a1, a2, a3b) that together with (91) and (93) imply Identities (89) and (90). Lemma 3. If (G, τ) is a topological quasigroup, then the functions t(a1, a2, a3) and p(a1, a2, a3) are jointly continuous in a1, a2, a3 in G. Proof. This follows immediately from Formulas (91), (93) and Deﬁni- tions 1, 6.

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