# SELECTED PROBLEMS AND RESULTS OF TOPOLOGICAL ALGEBRA

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## SELECTED PROBLEMS AND RESULTSOF TOPOLOGICAL ALGEBRA

Mitrofan M. Choban, Liubomir L. Chiriac

Tiraspol State University, Chi¸sin˘au, Republic of Moldova [email protected], [email protected]

Abstract In the class of topological algebras of a given signature the notion of totally bounded- ness and distinct notions of compactness are studied. The general properties of free topological algebras and compactifications of topological algebras are investigated too.

We discuss some old and new results and open problems.

Keywords:topological universal algebra, totally bounded algebra, pseudocompact space, Mal’cev al- gebra, homogeneous algebra, pseudocompact space, rezolvable space.

2010 MSC:08A05, 22A20, 54A25, 08B05, 08C10, 54C40, 54H15.

### 1.INTRODUCTION

We use the terminology from [12, 53, 33]. LetN={0,1,2, ...}andEbe the discrete sum of topological spaces {En : n ∈ N}. We say that En is the space of symbols of n-ary operations on topological E-algebras. A topological universal algebra of signatureEor a topologicalE-algebra is a non-empty topological spaceGon which there are given the continuous mappings{enG :En×GnG:n∈N}. The mappings enG form the algebraical structure onG.

LetG be a topological E-algebra, n ∈ Nand uEn. If n = 0, thenu(G0) = e0G({0} ×G0) is a singleton andu:G0Gis a mapping. Ifn≥1, then we consider then-ary operationu:GnG, whereu(x1, ...,xn)=enG(u,x1, ...,xn).

The polynomials are constructed in the following way:

-Eare polynomials;

- ifnN,n ≥ 1,uEn,pi is anmi-ary polynomial, then p = u(p1, ...,pn) is an m-ary polynomial, where

m=m1+m2+...+mnand

p(x1, ...,xm)=u(p1(x1, ...,xm1), ...,pn(xmn1+1, ...,xm)).

Let nm ≥ 1, p be an n-ary polynomial and q : {1,2, ...,n} → {1, ...,m} be a mapping. Thenv(x1, ...,xm) = p(xq(1),xq(2), ...,xq(n)) is anm-ary term. The polyno- mials are terms too. Ifuis ann-ary term andvis anm-ary term, thenu(x1, ...,xn)= v(y1, ...,ym) is an identity onE-algebras.

Denote by|X| the cardinality of the set X. Any space is considered to be aT1- space.

Leti∈ {−1,0,1,2,3,312}.

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A classKof topologicalE-algebra is called aTi-quasivarietyif:

- any algebraG∈Kis aTi-space,

- ifG∈KandBis a subalgebra ofG, thenB∈K,

- the topological product of algebras fromKis a topological algebra fromK, - if (G,T) ∈ K,T is aTi-topology onG and (G,T) is a topologicalE-algebra, then (G,T)∈K.

IfΩis a set of identities and V(E,Ω,i) is the class of all topologicalE-algebras with identitiesΦ, which areTi-spaces, thenV(E,Ω,i) is aTi-variety. AnyTi-variety is aTi-quasivariety.

A classVofE-algebras is non-trivial if|G| ≥2 for someGV.

The investigations of topological algebras are eﬀected in the following directions.

DP.Investigation of the relationship between the algebraic and topological prop- erties of the topological E-algebras G from V(E,Ω,i).

The afore named ProblemDPis examined in light of the following problems.

DT. Let G be an E-algebra. Determine the kinds of topologies, which can be considered on the E-algebra G that makes it a topological E-algebra.

DA.Let G be a topological space. Determine the types of algebraic structures that can be considered on the space G, which makes it a topological E-algebra.

DC.Application of the Theory of Topological Algebras.

### 2.COMPATIBILITY AND INCOMPATIBILITY

Fix a signatureE = ⊕{En : nN} and a setΩof identities. One of the general problems, determined by the direction DA, is the next.

Problem 2.1. Let G be a topological non-empty space, E be a signature andbe a set of identities. Is it true that G admits a structure of topological E-algebra for which GV(E,Ω,−1)?

One of the first results in this direction is the Pontryagin variant of the Frobenius theorem in the abstract algebra (see [89, 90]).

Theorem 2.1. (Frobenius - Pontryagin). Let D be a connected locally compact divi- sion ring. Then:

1. If D is associative and commutative, then either D is the ring of realsR, or the ringCof complex numbers.

2. If D is associative and non-commutative, then D is the ring of quaternionsH. 3. If D is non-associative, then D is the ring of octonionsD.

The algebra of quaternions was discovered by Hamilton in 1843 and the algebra of the octonions - by J. T. Graves in 1843. The Cayley-Diskson construction produces a sequence of topological algebras over the given topological field (in particular over the reals). In the case of reals, we obtain the algebrasR,C,H,D(see [14]).

Really, letRbe a topological ring with involutionxx. Denote byA(R,∗) the setR2 =R×Rwith the operations:

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(x,y)+(u,v)=(x+u,y+v);

(x,y)·(u,v)=(xu−vy,vx+yu);

(x,y)=(x,−y).

ThenA(R,∗) is a topological ring with the involution and a topologicalR-module.

The mapping x → (x,0) is the natural embedding of the ringRinto A(R,∗). As a rule, the point xRis identified by the point (x,0)∈A(R,∗) and one may consider thatRA(R,∗).

If on the fieldRof reals the identical mappingxx= xis the given involution, then C= A(R,∗) is the algebra of complex numbers,H= A(C,∗) is the algebra of quaternions (hypercomplex) number and D = A(H,∗) is the algebra of octonions.

The algebras H1 = A(H,∗) and Hn+1 = A(Hn,∗) relatively to the multiplication are not with division for alln.

Corollary 2.1. Let G be an infinite connected and locally compact space. If dim G<{1,2,4,8}, then G does not admit the structure of the topological division ring.

Obviously, any topological spaceGadmits structures of topological E-algebras.

For this it is suﬃcient to fix some continuous mapping enG : En ×GnG for any nN. In particular, the operation xy = x determines onGthe structure of a topological semigroup with a right identity: the elementeGis a right (respectively, left) identity ifxe=x(respectively,ex=x) for anyxG.

Remark 2.1. There exists a metrizable connected compact space A such that if xy is a structure of a topological groupoid with right identity, then xy= x for all x,yA.

In this case any continuous mappingφ : A×AA is one of the projections or a constant mapping. The space A is called the Cook continuum (see [89, 90]).

Theorem 2.2. (L. M. James, [63, 64]) If n <{0,1,3,7}, then on the sphere Sn from the(n+1)-dimensional Euclidean space En+1does not exist the structure of a topo- logical groupoid xy with the identity eSn.

Theorem of L.M.James and the fixed point principle have many applications.

Corollary 2.2. Let n ≥ 1, Bn = {xEn : ∥x∥ ≤ 1}, and eSn1BnEn. The following assertions are equivalent:

1. On the sphere Sn1there exists the structure xy of a topological groupoid with the identity eSn1.

2. On Euclidean space En there exists the structure xy of a topological groupoid with the identity eSn−1such that Sn−1and Bnare subgroupoids.

3. On Euclidean space En there exists the structure xy of a topological groupoid with the identity eSn1such that Bn\ {xy:x,ySn1},∅.

4. n∈ {1,2,4,8}

Proof. Implications 1→4→1 immediately follows from the James’ Theorem 2.2.

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Assume thatx·yis a structure of a topological groupoid onSn−1with the identity eSn1. Let 0 = (0, ...,0) be the neutral element of the Euclidean space En. If xEnandx,0, then there exists a unique pointp(x)Sn−1such thatp(x)= 1xx.

The mappingh : En\ {0} −→ Sn−1 is continuous. Now we put x∗0 = 0∗x = 0 for eachxEn andyz = ∥y∥ · ∥z∥ ·h(y)·h(z) for ally,zEn\ {0}. Then (En,∗) is a topological groupoid with the identity e and (Sn1,·), (Bn,∗) are subroupoids.

Obviously x·y = xy for x,ySn1. Implication 1 → 2 is proved. Implication 2→3 is obvious.

Assume thatxyis a structure of a topological groupoid onEnwith the identityeSn1andBn\ {xy:x,ySn1},∅. We can suppose that 0∈Bn\ {xy:x,ySn1}. Thenxy= h(xy) is a structure of a topological groupoid onSn−1with the identity eSn1. Implication 3→1 is proved. The proof is complete.

We need some definitions. A topological quasigroup is a non-empty spaceGwith three binary operations {·,r,l} and identities x · l(x,y) = r(y,x) · x = l(x,x ·y) = l(r(x,y)·x))=r(y·x,x)=y.

A homogeneous algebra is a non-empty spaceGwith two binary operations{+,·}

and the identitiesx+x·y=x·(x+y)=y,x·x=y·y.

A biternary Mal’cev [72] algebra is a non-empty space with two ternary operations {p,q}and identitiesp(y,y,x)=q(p(x,y,z),y,z)= p(q(x,y,z),y,z)=x.

A Mal’cev algebra is a non-empty space with one ternary operation{p}and iden- titiesp(x,x,y)=p(y,x,x)=y.

A topological quasigroup with the identity is a loop. Every topological group is a loop. A space admits a structure of a topological quasigroup if an only if it admits a structure of a topological loop (A. I. Mal’cev, 1956, [72]). Any biternary Mal’cev algebra is a Mal’cev algebra (A. I. Mal’cev, 1956, [72]). Any topological quasigroup admits a structure of a biternary Mal’cev algebra (A. I. Mal’cev, 1956, [72]). A space admits a structure of a homogeneous algebra if and only if admits a structure of a biternary Mal’cev algebra (M. M. Choban [28]). A space X admits a structure of a homogeneous algebra if and only if X is a rectifiable space, i.e. there exist a homeomorphismh: X×XX×Xand a pointcXsuch thath(x×X) = x×X and h(x,x) = (x,c) for any xX (M. Choban [28]). The mapping h is called a rectification onX.

A spaceXis homogeneous if for any two pointsa,bXthere exists a homeomor- phismhab :XXsuch thathab(a)=b.

Let{+,·}be a structure of a homogeneous algebra on a spaceG,a,bGandx·y

=c for all xG. ThenPa(x) = a·x, Qa(u) = a+xare homeomorphisms, Pa1= Qa,Pa(a)= c andQa(c) = a. OnG there exists a structure {+,·} of homogeneous algebra such thatcis the a priori given point. The mapping Ψ(x,y) =(x,x·y) is a homeomorphism ofG×GontoG×Gsuch thatΨ(x,x) =(x,c) andΨ({x} ×G)= {x} ×Gfor anyxG.

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Assume now thatcX andh :G×GG×Gis a homeomorphism such that h(x,x)=(x,c) andh({x} ×G) ={x} ×Gfor any xG. Let p :G×GGbe the projectionp(x,y)=y. We putp(h(x,y))=x·yandp(h1(x,y))=x+yfor allx,yG.

Then{+,·}is a structure of a homogeneous algebra on a spaceG.

Let now fb(x)= x+b. Sinceh−1(G×c)={(x,x) :xG}andh−1(G× {b}) is the graphic of the mapping fb, then forb , c we have fb(x) , x for anyxG. Thus the mapping fbdoes not contains fixed points for anyb,c. In particular,Gis not a fixed point space. This simple fact was observed by A. S. Gul’ko ([58], Proposition 4.1). From this fact it follows.

Corollary 2.3. Any homogeneous algebra G is a homogeneous space. If |G| ≥ 2, then G is not a fixed point space.

Let XY. The mappingr : YX is a retraction ifr(x) = x for all xX.

If p : Y3Y is a Mal’cev ternary operation onY, thenq(x,y,z) = r(p(x,y,z)) is a ternary Mal’cev operation onX. Thus a retract of a Mal’cev algebra is a Mal’cev algebra. In particular, anyAR-space admits a structure of a Mal’cev algebra.

Corollary 2.4. For any cardinal τ ≥ 1 the cube Iτ is a Mal’cev algebra and it does not admit a structure of homogeneous algebra. For τinfinite the space Iτ is homogeneous.

Corollary 2.5. Any AR-space is a fixed point space, admits a structure of a Mal’cev space and does not admit a structure of a homogeneous algebra.

If a compact spaceXadmits a structure of a Mal’cev algebra, thenXis a Dugundji space (see [31, 32, 33, 79, 95]). In [8] it was proved that for a Hausdorﬀcompact- ification bX of a rectifiable space X the remainder bX\X is a pseudocompact or a Lindel¨of space. The last assertion is not true for Mal’cev algebras [8].

The next questions are open.

Problem 2.2. Is it true that any Mal’cev algebra is a retract of some homogeneous algebra, or of some topological quasigroup?

Problem 2.3. (A.V.Arhangel’skii). Is it true that any compact Mal’cev algebra is a retract of some compact group?

Problem 2.4. Let X be a first-countable completely regular space, the Souslin num- ber c(Xτ)is countable for any cardinal τand Xm admits a structure of a homoge- neous algebra for some cardinal m. Is it true that the space X0 admits a structure of a homogeneous algebra?

The minimal infinite cardinal numberτfor which|γ| ≤ τfor any disjoint family γ of open subsets of a spaceX is called theSouslin number of the space X and it is noted byc(X).

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Remark 2.2. Let X be a space and A be a non-empty set. Fix a point0 ∈ X and an elementα ∈ A. For any xX we put e(x) = (xβ : β ∈ A)XA, where xα = x and xβ = 0for all β , α. Then h : X −→ XA is an embedding. We identify X and h(X) and consider that X = h(X)XA. Then the mapping r : XA −→ X, where r(xβ :β∈A)=h(xα), is a retraction. Thus the following assertions are equivalent:

1. The space X admits a structure of a Mal’cev algebra.

2. The space Xτadmits a structure of a Mal’cev algebra for any cardinal number τ.

3. The space Xτadmits a structure of a Mal’cev algebra for some cardinal number τ≥1.

### 3.PRECOMPACT TOPOLOGIES ON ALGEBRAS

Fix a discrete signatureE=⊕{En:nN}. A topologicalE-algebraGis precom- pact ifG is a topological E-algebra of some Hausdorﬀcompact E-algebra. In this section any space is considered to be completely regular.

LetG be a topological E-algebra. A pair (B, φ) is an a-compactification or an almost periodic compactification ofGif Bis a compactE-algebra,φ :GBis a continuous homomorphism and the setφ(G) is dense inB.

If (B, φ) and (H, ψ) area-compactifications ofG, then (H, ψ)≤(B, φ) if there exists a continuous homomorphismg : BH such thatψ=g◦φ. For any topological E-algebra the classAC(G) of alla-compactifications ofGis a complete lattice. The maximala-compactification (bhG,bG) ofGis called the Bohr-Holm compactification ofG. The mappingbG :GbhGis an embedding if and only ifGis precompact.

The Bohr-Holm compactifications were studied in [62, 60, 61, 37, 38, 42, 43, 76, 86].

LetGbe a topologicalE-algebra andGdbe the algebraGwith the discrete topol- ogy. A pair (H, φ) is called anap-extension ofGif (H, φ) is ana-compactification of Gdand (bhG,bG)≤(H, φ). Thus the classEP(G) of allap-extensions ofGis a com- plete lattice with the maximal element (apG,aG) and minimal element (bhG,bG). If the spaceGis discrete, thenapG=bhG.

Let C be the field of complex numbers and C(X) be the Banach algebra of all continuous bonded complex-valued functions on the space X. By B(X) denote the Banach-algebra of bounded Baire-measurable complex-valued functions onX. The algebraBa(X) of Baire-measurable sets of the spaceXis theσ-algebra generated by the class of functionally closed sets{f1(0) : fC(X)}of the spaceX. A function g : X −→ Cis Baire-measurable ifg1(U) ∈ Ba(X) for each open subset U ofC. The algebra of functional-measurable sets Fun(X) of the space X is theσ-algebra generated by the class of functionally sets{f1(H) : fC(X),H ⊆ C}of the space X. A functiong :X −→Cis functionally-measurable ifg1(U) ∈ Fun(X) for each open subset U ⊆ C. By Φ(X) denote the Banach-algebra of bounded functional- measurable complex-valued functions on X. ByF(X) denote the Banach-algebra of

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all bounded complex-valued functions on X. Obviously,C(X)B(X) ⊆ Φ(X) ⊆ F(X).

If G is a topological E-algebra and (H, φ) is an a-compactification of G, then APC(H,φ)(G)={f ◦φ: fC(H)}. If (H, φ) is anap-extension ofG, thenAP(H,φ)(G)

= {f ◦φ : fC(H)}. Let AP(G)= AP(apG,aG)(G) and APC(G) = APC(bhG,bg)(G).

ThenAP(G) is the Banach algebra of all almost periodic functions onGandAPC(G) is the Banach algebra of all almost periodic continuous functions onG.

IfGis a topological group, then the function fF(G) is almost periodic if the closure of the set{fa : aG}, where fa(x) = f(ax) for all a,xG, in F(G) is a compact set.

Remark 3.1. For a subalgebra LAP(G)the following assertions are equivalent:

AP1. L= AP(H)(G)for some ap-extension(H, φ)of G.

AP2.The algebra L has the next properties:

- APC(G)L;

- L is closed in AP(G);

- if fL, then fL.

Theorem 3.1. Let X be a pseudocompact space. Then there exists a one-to-one mappingΨ:Φ(βX)→Φ(X)with the properties:

1.Ψ(f)= f|X andf∥=∥Ψ(f)∥.

2.Ψ(f +g)= Ψ(f)+ Ψ(g)andΨ(f ·g)= Ψ(f)·Ψ(g).

3. If the sequence {fn ∈ Φ(βX) : nN} converges pointwise to the function fF(X), then f ∈Φ(βX)and the sequence{Ψ(fn) : nN}converges pointwise to Ψ(f).

4.Ψ(C(βX)=C(X),Ψ(B(βX)=B(X)andΨ(Φ(βX)=Φ(X).

5. If X is a topological group, then the function fB(βX)is almost periodic on βX if and only if the functionΨ(f)is almost periodic on X.

Proof. Assertions 1 - 4 were proved in [27]. Really, for any bounded continuous function fC(X) there exists a unique continuous function βf on βX such that f = βf|X. Thus for each functionally-measurable setLof the space X there exists a functionally-measurable setLβof the spaceβX such thatL = LβX. For the set Lβ and any point xLβ there exists a Gδ-subset E ofβX such that xELβ. Hence, since the spaceXis pseudocompact, the setLβis unique. Therefore, for each functiong ∈ Φ(X) there exists a unique function βg ∈ Φ(βX) such that g = βg|X and the operatorΨ(f)= f|Xis a one-to-one mapping ofΦ(βX) ontoΦ(X). This fact proves the assertions 1 - 4. Assertion 5 is obvious. The proof is complete.

Let G be a pseudocompact E-algebra. If (H, φ) is an ap-extension ofG, then denote by GH = φ(G) the algebra G as a topological subalgebra of the compact algebra H. Theap-extension (H, φ) is called B-measurable if APHB(G). The ap-extension (H, φ) is calledap-pseudocompact if the spaceGHis pseudocompact.

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Theorem 3.2. Let G be a pseudocompact group and(H, φ)be an ap-pseudo-compact ap-extension of G. Then AP(H)(G)∩Φ(G)=C(G).

Proof. Let βG be the Stone- ˇCech compactification of the pseudocompact group.

ThenβGis a topological group andGbe a dense subgroup ofβG(see [12]). There ex- ists a continuous homomorphismϕ:H−→βGsuch thatϕ(x)=xfor anyxGH.

Assume that f ∈ (AP(H)(G)∩Φ(G)). Then, by virtue of Theorem 3.1, there exist g ∈Φ(βG)) andg1C(H) such that f = g|Gandg(ϕ(z))=g1(z) for eachzH. If Bis a closed subset ofC, then, since the functiong1is continuous, the setg−11 (B) is closed inH. Since the mappingϕis closed, the setϕ(g−11 (B))= g−1(B) is closed in βG. Hence the functionsgand f are continuous. The proof is complete.

Therefore the almost periodicity of the functional-measurable function is in op- posite with pseudocompactness. In this context it is interesting to mention the next three results.

Theorem 3.3. (P. Kirku [70]) Let G be a divisible torsion-free Abelian group of the uncountable cardinality|G|=2α. Then G admits exactly2τ-many compact group topologies.

Theorem 3.4. (W. W. Comfort and D. Remus [48, 46]). Let (G,T) be a compact Abelian group. Then G has a pseudocompact group topology WT such that the weight w(G,W)≥2w(G,T).

Existence of compact and pseudocompact topologies on groups and rings were studied in [44, 68, 69, 94].

Let (G,T) be a compact group andWbe a pseudocompact group topology onG such thatTW. Then the Stone- ˇCech compactificationHof the group (G,W) is an ap-pseudocompactap-compactification of the groupG.

Theorem 3.5. (W. Comfort, S. U. Ruczkowski and F. J. Trigos-Arrieta [47]). Every infinite Abelian group G admits a familyAof totally bounded group topologies with

|A| =22|G| and the spaces(G,T),(G,w)are not homeomorphic for distinct(T,W) ∈ A.

A cardinal numberτis a strong limit cardinal if 2m< τprovidedm< τ. By virtue of Theorem 9.11.2 from ([12], p. 672) it follows:

Corollary 3.1. Letτbe a sequential strong limit cardinal. Then no group of cardi- nalityτadmits a pseudocompact group topology.

There exist many sequential strong limit cardinals. Letτ ≥ 2. We put 1(τ) = 2τ,(n+1)(τ)= 2n(τ)andω(τ)= sup{n(τ) :nN}. Thenω(τ) is a sequential strong limit cardinal.

Under Martin’s Axiom MA the infinite Abelian groupGadmits a pseudocompact group topology if and only ifGadmits a countable compact group topology without

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non-trivial convergent sequence. ([12], Theorem 9.12.9, D. Dikranjan and M. G.

Tkachenko).

The following questions are intriguing.

Problem 3.1. Let G admits some totally bounded topology and consider G as a sub- space of the space apG.

a. Is it true that any bounded subset of G is finite?

b. Is G as a subspace of apG a Dieudonn´e complete space?

c. Is G closed in apG relatively to the Gδ-topology on apG?

A spaceXis Dieudonn´e complete if it is complete relatively to the maximal uni- formity. A subsetLof a spaceXis bounded if any continuous function f : XR is bounded onL. For Abelian groups the answer to the question in Problem 3.1.a is

”Yes” ([12],Theorem 9.9.42 of F. J. Trigos-Arrieta). The finiteness of compact sub- setsFGofapGfor AbelianGwas established by H. Leptin [71] and I. Glicksberg [55].

IfHis a measurable subgroup of the compact groupGwith the Haar measureλ, then orHis open inGorλ(H)=0. Letλbe the Haar measure onapG, whereGis a group with some precompact topology. Then orλ(G) =0, orGis not measurable in apGandλ(U)=1 for any measurable setUofapGwhich containsG. For example λ(G)=0, if|G|<20, andλ(U)=1 for any measurable setUofapGwhich contains G, ifGadmits pseudocompact group topologies. Under which conditionsλ(G)=0?

### SEMITOPOLOGICAL ALGEBRAS

Fix a discrete signatureE = ⊕{En :nN}and the subspacesSE andPE.

AnE-algebraGwith the topologyT is called:

- anS-semitopologicalE-algebra if the operationu :GnGis separately con- tinuous for allnNanduSEn;

- aP-paratopologicalE-algebra if the operationu:GnGis continuous for all nNanduPEn;

- a (P,S)-quasitopological E-algebra if G is an S-semitopological and a P-paratopologicalE-algebra.

Any P-paratopological E-algebra is a topologicalP-algebra. In natural way the notion of aTi-quasivariety of (P,S)-quasitopologicalE-algebra is defined.

Theorem 4.1. (M. Choban [28]) Let V be a Ti-quasivariety of(P,S)-quasitopological E-algebras. Then for any non-empty space X there exists an algebra F(X,V)V and a continuous mappingφX :XF(X,V)such that:

1. The setφX(X)algebraically generates the E-algebra F⇁(X,V).

2. For any continuous mapping g : XGV there exists a continuous homo- morphism g:F(X,V)→G such that g=g◦φX.

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The pair (F(X,V), φX) is called a free (P,S)-quasitopological E-algebra of the spaceXin the classV.

The algebraF(X,V) is abstract free if for any mappingg:XGVthere exists a homomorphismg:F(X,V)Gsuch thatg=g◦φX.

Problem 4.1. Assume that there exists a space GV with a proper open subset.

a. Under which conditions the mappingφX :XF(X,V)is an embedding?

b. Under which conditions the algebra F(X,V)is abstract free?

For varieties of topologicalE-algebras the Problems 4.1 were formulated by A. I.

Mal’cev [72]. The answers are positive for any completely regular Hausdorﬀspace [28].

Let{·,1,e}be the signature of groups. IfS = P= {·}then anS-semitopological group is called a semitopological group and a P-paratopological group is called a paratopological group.

LetZbe the discrete group of integers.

If V is aTi-quasivariety of semitopological groups and Vp = {GV : G is a paratopological group},Vg={G∈V :G is a topological group}, then:

1.VgVpV;

2. IfGVandGdis the groupGwith the discrete topology, thenGdVg; 3. If (F(X,V), φX),(F(X,Vp), φpX) and (F(X,Vg), φgX) are the free objects of a space X, then there exist the continuous homomorphismsψX :F(X,V)F(X,Vp) andθX :F(X,Vp)→F(X,Vg) such thatφpXX ◦φX andφgXX◦φpX;

4. For any completely regular space X the mappings ψX and θX are continuous isomorphisms.

Theorem 4.2. Let i∈ {−1,0,1,312}, V be a Ti-quasivariety of semitopological groups andZ∈V. Then for any Ti-space X:

1.φX :XF(X,V)andφpX:XF(X,Vp)are embeddings.

2. The groups F(X,V)and F(X,Vp)are abstract free in V and Vprespectively.

Proof. Consider the following four cases.

Case 1.i=312.

This case was proved in [28].

Case 2.i=1.

On any setXconsider the cofinite topologyTc f ={X} ∪ {X\F :F is a finite set}. Then (X,Tc f) is a compact T1-space. IfG is a group, then (G,Tc f) is a semitopo- logical compact group. We can assume thatX = φXd(X) ⊆ F(Xd,V) as a set. Fix a T1-spaceX. The groupF(Xd,V) is the abstract free group of the setXin the classV. Since (F(Xd,V),Tc f) ∈ V, there exists a unique continuous homomorphism g : F(X,V) −→ (F(Xd,V),Tc f) such that g(φX(x)) = xfor each xX. Theng is an isomorphism and the object F(X,V) is abstract free inV. Obviously, thatφX is an embedding for the space (X,Tc f). Since anyT1-spaceXfor some cardinal numberτ admits an embedding in (F(Xd,V),Tc f)τ, the mappingφXis an embedding.

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Case 3.i=0.

LetDω be the groupZwith the topology{∅} ∪ {Un = {m ∈ Z :mn} : n ∈ }. ThenDωVpV. Leta,bbe two distinct points of aT0-spaceX. Assume thatU is open inX,a < UandbU. Then the mappingg :XDω, whereg1(1) = U andg−1(0)= X\U is continuous. ThusφpX :XF(X,Vp) is an embedding. The assertion 1 is proved. The proof of the assertion 2 is proved in [36].

Case 4.i=−1.

LetXbe a space. LetG0 be the groupZ×Zwith the topology{∅} ∪ {Vn = {m∈ Z : mn} ×Z : n ∈ Z}. The space Xadmits an embedding inGw(X)0 . ThusφX is an embedding and we can assume thatX = φX(X) ⊆ F(X,V). LetGX be the group F(Xd,V) with the anti-discrete topology{∅,F(Xd,V)}andXXd(X)⊆ F(Xd,V) as a set. Then the identical mapping f : X −→GX, where f(x) = xfor eachxX is a continuous mapping and there exists a continuous homomorphismg:F(X,V)−→

GX such that f = g|X. Sincegis an isomorphism, the groupF(X,V) is abstract free in the classV.

The proof is complete.

Theorem 4.3. Let i ∈ {1,312}, V be a non-trivial Ti-quasivariety of semitopological groups andZ<V. Then for any Ti-space X:

1.φX :XF(X,V)andφpX:XF(X,Vp)are embeddings.

2. The groups F(X,V)and F(X,Vp)are abstract free in V and Vprespectively.

Proof. Consider the following two cases.

Case 1.i=312.

This case is proved in [28].

Case 2.i=1.

This case is similar to the case 2 in the proof of the previous theorem.

A groupG with a topology is called aleft(respectively,right) topological group if the left translation La(x) = ax (respectively, the right translationRa(x) = xa) is continuous for anyaG.

A classVof left topological groups is called aTi-quasivariety of left topological groups if:

(LF1) the classVis multiplicative;

(LF2) ifG∈VandAis a subgroup ofG, thenA∈V; (LF3) every spaceG∈Vis aTi-space;

(LF4) ifG ∈V,Tis a compactTi-topology onGand (G,T) is a left topological group, then (G,T)∈V;

From Theorems 4.2 and 4.3 it follows

Corollary 4.1. Let i∈ {−1,0,1,312}, V be a Ti-quasivariety of left topological groups andZ∈V. Then for any Ti-space X:

1.φX :XF(X,V)andφpX:XF(X,Vp)are embeddings.

2. The groups F(X,V)and F(X,Vp)are abstract free in V and Vprespectively.

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Corollary 4.2. Let i ∈ {1,312}, V be a non-trivial Ti-quasivariety of left topological groups andZ∈V. Then for any Ti-space X:

1.φX :XF(X,V)andφpX:XF(X,Vp)are embeddings.

2. The groups F(X,V)and F(X,Vp)are abstract free in V and Vprespectively.

The following assertion completes Theorem 4.3 and Corollary 4.2.

Lemma 4.1. Let G be a left topological group and for any xG there exists n(x)∈N such that xn(x)=e. Then G is a T1-space.

Proof. Any finiteT0-space contains a closed one-point subset. Thus any finite left topological group is aT1-space. By hypothesis, any pointaGis contained in the finite subgroupG(a) ={ai : 0≤in(a)}. Thus{e}is a closed subset of the groupG andGis aT1-space.

Remark 4.1. The similar assertions are true for classes of right topological groups.

Remark 4.2. Let V be the class of all paratopological groups, or of all paratopo- logical Abelian groups. In [88] it was proved that the answers to the questions from Problems 4.1 are positive for any T0-space X. For this the authors of [88] use the method of left (right) invariant pseudo-quasi-metrics. Since topology generated by the left (right) invariant pseudo-quasi-metrics may not be a paratopological topol- ogy [74, 12, 17], this point of view may create dangerous moments. Nevertheless, the extensions of the quasi-metrics from [88] are invariant quasi-metrics. For this in [36] we use the method of invariant pseudo-quasi-metrics. The method of left (right) invariant pseudo-metrics was proposed in [74] and [17]. The method of invariant pseudo-metrics on free objects was developed in [57, 30].

LetSE,Gbe anE-algebra,n≥1, j∈ {1,2, ...,n},uEnS anda1,a2, ...,anG. We putR(G,j,u,a1, ...,an)={xG:u(a1, ...,aj1,x,aj+1, ...,an)=aj}.

The E-algebraGis called anS-simpleE-algebra if for alln ≥ 1, j∈ {1,2, ...,n}, uEnS and a1,a2, ...,anG we have R(G,j,u,a1, ...,an) = G or the set R(G, j,u,a1, ...,an) is finite.

All quasigroups are simple algebras.

Theorem 4.4. Let SE, i ∈ {−1,0,1}and V be a non-trivial Ti-quasivariety of S -semitopological S -simple E-algebras. Then for any T1-space X:

1. the mappingφX :XF(X,V)is an embedding.

2. the algebra F(X,V)is abstract free.

Proof. LetGV. Denote by Tc f = {∅} ∪ {G\F : F is a finite set} the co-finite topology onG. SinceGis anS-semitopologicalS-simpleE-algebra the operation u:GnGis separately continuous for allnNanduSEn. Thus (G,Tc f)∈V. Fix a non-emptyT1-spaceX. Denote byXdthe set Xwith the discrete topology.

Then the E-algebra (F(Xd,V), φXd) is the abstract free algebra of the spaceXin the

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classV. LetGX be the algebraF(Xd,V) with the co-finite topologyTc f. ThenGXV, the mapping g = φXd : X −→GX is continuous and an injection. There exists a continuous homomorphismh : F(X,V) −→ GX such that h(φX(x)) = g(x) for each xX. Henceg is an isomorphism and the algebra F(X,V) is abstract free. Since

|X| ≤ |GX|, then for some cardinalτthe space Xadmits an embedding inGτX. Thus the mappingφX :XF(X,V) is an embedding. The proof is complete

Now we mention the following open problems.

Problem 4.2. a. Let i∈ {2,3}and V be a non-trivial Ti-quasivariety of semitopolog- ical groups. Are Theorems 4.2 and 4.3 true?

b. Let i ∈ {2,3}and V be a non-trivial Ti-quasivariety of left topological groups.

Are Corollaries 4.1 and 4.6 true?

### 5.THEOREMS OF MONTGOMERY AND ELLIS

In 1936 D. Montgomery [75] set the following problems.

Problem 1G.Under which conditions a semitopological group is a paratopologi- cal group?

Problem 2G. Under which conditions a paratopological group is a topological group?

D. Montgomery [75] has proved that every complete matrizable semitopological group is a paratopological group and every complete metrizable separable semitopo- logical group is a topological group. In 1957 R. Ellis (see [52, 12]) showed that any locally compact semitopological group is a topological group.

In 1960, W. Zelazko [100] established that any complete metrizable semitopolog- ical group is a topological group. Then in 1982 N. Brand [22] proved that a ˇCech complete paratopological groups is a topological group. A. Bouziad [19, 20, 21]

proved this assertion for semitopological groups. Many interesting results were ob- tained in [7, 6, 10, 65, 23, 54, 59, 80, 83, 87, 97].

We mention the following two result.

Theorem 5.1. ( P. Kenderov, I. S. Kortezov and W. B. Moors [65, 10]) If a regu- lar semitopological group G contains a dense ˇCeeh complete subspace, then G is a topological group.

Theorem 5.2. (A. Arhangelskii and M. M. Choban [6, 7, 10]) If a regular paratopo- logical group G contains a dense subspace which is a dense Gδ-subspace of some pseudocompact space, then G is a topological group and a dense Gδ-subspace of some pseudocompact space.

Let{·,r,l}be the signature of quasigroups.

A quasigroupGwith a topology is called:

- a paratopological quasigroup if the multiplicative operation{·} and the transla- tionsla=l(a,x),ra=r(x,a),aG, are continuous;

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- a semitopological quasigroup if the translations La(x) = a· x, Ra(x) = x ·a, la=l(a,x),ra=r(x,a),aG, are continuous.

Any paratopological (respectively, semitopological) group is a paratopological (re- spectively, semitopological) group. Any paratopological quasigroup is a semitopo- logical quasigroup. In a semitopological quasigroup all translations La(x) = a·x, Ra(x) = x·a, la = l(a,x),ra = r(x,a),aG, are homeomorphisms. Moreover, la=La1andra=Ra1for eachaG.

The next problems are similar to the Montgomery’s problems.

Problem 5.1. Under which conditions a semitopological quasigroup is a paratopo- logical quasigroup?

Problem 5.2. Under which conditions a paratopological quasigroup is a topological quasigroup?

Let (G,·) be a groupoid. Denote by P(G,·) the minimal semigroup of mappings g:G−→Gsuch thatLa,RaP(G,·) for eachaG.

AT-groupoid (or a Toyoda groupoid) is a non-empty setGwith one binary oper- ation{·}and four unary operations{a1,a2,b1,b2}such that:

ifxy=a1(x)·b1(y)), then (G,◦) is a group;

a1(a2(x))=b1(b2(x))=xfor eachxG;

{a1,a2} ∩P(G,·),∅and{b1,b2} ∩P(G,·),∅.

In this case we say that (G,◦) is the group associated to the T-groupoid (G,·,a1,a2,b1,b2). By definitions,a2 =a11andb2=b11.

AnyT-groupoid is a quasigroup.

Let (G,◦) be the topological group associated to a topological T-groupoid (G,·,a1,a2,b1,b2). By virtue of Albert’s theorem [2, 3], all topological groups (G,◦) associated to the given T-groupoid are topologically isomorphic. In this sens that group is unique. Hence, if the topological quasigroup (G,·,r,l) for some mappings {a1,a2,b1,b2,c1,c2}is a topologicalT-groupoid, then:

- we havex·y=a2(x)◦b2(y),l(x,y)=b1(a2(x)1y) andr(x,y)=a1(x◦b2(y)1);

- there exists many structures of the kind{a1,a2,b1,b2}onG;

- all topological groups associated to theT-groupoids (G,·,a1,a2,b1,b2) are topo- logically isomorphic.

Therefore any topologicalT-groupoid is considered a topological quasigroup, too.

Moreover,we assume that the T-groupoid (G,·) as a universal algebra is the quasi- group (G,·,r,l). Distinct classes ofT-quasigroups were introduced and studied in [66, 67, 15, 16, 41]. For this general case we use the notion of a ”T-groupoid”. Since any Hausdotﬀ topological group is a completely regular space, then the space of a topologicalT-groupoid is completely regular provided it is aT0-space.

AT-groupoid (G,·,a1,a2,b1,b2) with a topology is called:

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- a topologicalT-groupoid if the operation (G,·,a1,a2,b1,b2) are continuous and Gis a topological quasigroup;

- a paratopologicalT-groupoid if the operation{·,a1,a2,b1,b2}are continuous;

- a semitopologicalT-groupoid if the multiplicative operation{·}is separate con- tinuous and the operation{a1,a2,b1,b2}are continuous.

If a (G,·,a1,a2,b1,b2) is a semitopological T-groupoid, then the operations {a1,a2,b1,b2}are homeomorphisms. Moreover, if a T-groupoid (G,·,a1,a2,b1,b2) is a semitopological quasigroup, then the operation {a1,a2,b1,b2} are homeomor- phisms.

We mention that aT-groupoid (G,·,a1,a2,b1,b2) with topology:

- is a topologicalT-groupoid if and only if (G,·,r,l) (Gr,l) is a topological quasi- group;

- is a paratopologicalT-groupoid if and only if (G,·,r,l) (Gr,l) is a paratopolog- ical quasigroup;

- is a semitopological T-groupoid if and only if (G,·,r,l) (G,·r,l) is a semitopo- logical quasigroup.

Any group with the identical mappings{a1,a2,b1,b2}is considered aT-groupoid too. Therefore:

- any semitopological group is a a semitopologicalT-groupoid;

- any paratopological group is a a paratopologicalT-groupoid;

- any topological group is a a topologicalT-groupoid.

By virtue of K. Toyoda theorem [93] it follows that:

- any medial quasigroup is aT-groupoid;

- any semitopological medial quasigroup is a a semitopologicalT-groupoid;

- any paratopological medial quasigroup is a a paratopologicalT-groupoid;

- any topological medial quasigroup is a a topologicalT-groupoid.

Theorem 5.3. LetKbe a class of topological spaces. Then:

1. Any semitopological T -groupoid G∈Kis a topological quasigroup if and only if any semitopological group H∈Kis a topological group.

2. Any paratopological T -groupoid G∈Kis a topological quasigroup if and only if any paratopological group H∈Kis a topological group.

Proof. LetH = (G,◦) be the associated group at theT-groupoid (G,·,a1,a2,b1,b2) with the topology and{a1,a2,b1,b2}.

Then:

-His a semitopological group if and only ifGis a semitopologicalT-groupoid ; -His a paratopological group if and only ifGis a paratopologicalT-groupoid;

-His a topological group if and only ifGis a topological quasigroup, i.e a topo- logicalT-groupoid.

The proof is complete.

Hence, Theorems 5.1 and 5.2 are true for medial quasigroups and for paramedial quasigroups.

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Problem 5.3. Is Theorem 5.2 true for any quasigroups? In particular, is Theorem 5.2 true for IP-quasigroups?

Problem 5.4. Is Theorem 5.2 true for any quasigroup? In particular, is Theorem 5.2 true for IP-quasigroups?

Distinct classes of spaces and algebras were studied in [5, 6, 8, 9, 10, 11, 13, 29, 18, 24, 45, 49, 73, 77, 78, 84, 85, 91, 96, 98, 99].

### 6.SOLVABILITY OF ALGEBRAS

Let X be a space and τ be a cardinal. The spaceX is called τ-solvable if there exists a family{Xα :α∈ A}of pairewise disjoint dense subspaces such that|A| ≥τ.

A 2-solvable space is called solvable. A|X|-solvable space is called totally solvable.

LetTbe a topology on a quasigroupG. The topologyTis weakly bounded if for any non-empty setU ∈Tthere exists a finite setLGsuch thatG = L·U. We do not suppose that (G,T) is a topological, or a semitopological quasigroup.

Example 6.1. Denote by T1(G)={X} ∪ {X\F :F is a finite subset of G}the minimal T1-topology on the quasigroup G, i.e. the cofinite topology on G. If bG, then T0(G,b)={UT1(G) :bU}is a T0-topology on G. Then:

- ifT⊆T1(G), thenTis a weakly bounded topology on G;

-(G,T1(G))is a semitopological quasigroup;

- if the set G is infinite, then(G,T1(G))is not a paratopological quasigroup;

- let G contains two distinct points and bG, then(G,T0(G,b))is not a semitopo- logical quasigroup.

Theorem 6.1. (M. Choban and L. Chiriac [39]) Let G be an infinite group of car- dinality τ. Then there exists a disjoint family {Bµ : µ ∈ M} of subsets of G such that:

1.|M|=|G|.

2. G=∪{Bµ :µ∈M}.

3.(G\Bµ)·K ,G for allµ∈M and every finite subset K of G.

4. The sets{Bµ :µ∈ M}are dense in all totally bounded topologies on G.

This theorem generalized a result of I. Protasov [81]. In [39] Theorem 6.1 is proved forIP-quasigroups. More general result was proved in [26].

Problem 6.1. Let G be a topological quasigroup (or IP-quasigroup). Is it true that G×G is a solvable space?

The answer is positive for groups (I. P. Protasov).

### 7.ON ALGEBRAS WITH DIVISIONS

LetEbe a signature. Ifn≥1,gEnand 1≤in, then

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g(a1, ...,ai−1,x,ai+1, ...,an)=ai is an equation onE-algebras. Denote bye(g,n,i) this equation.

Let φ be a set of equations on E-algebras. ByV(E, φ) we denote the class of all topological E-algebras on which the equationse(g,n,i) ∈ φ are solutions, i.e.

for any a1,a2, ...,anG there exists bG such that g(a1, ...,ai1,b,ai+1,an) = ai. By V(E,uφ) we denote the class of all algebras GV(E, φ) on which the equations e(g,n,i) ∈ φ are unique solutions. Lete(g,n,i) be an equation fromφ. We say that there exists a primitive solution on G of this equation if there exists a termh(y1,y2, ...,yn) such that g(a1, ...,ai−1,h(a1, ...,ai, ...,an),ai+1, ...,an) = ai for any a1,a2, ...,anG. LetV(E, φ,Π) be the class of E-algebrasGV(E, φ) with the primitive solutions for all equations fromφ. ObviouslyV(E, φ,Π)⊆ V(E, φ). In some cases we may extend the signatureEand consider that the solutions fromφare operations from the signature.

There existsE-algebras in which some equations are solutions but does not exist primitive continuous solutions. From this point of view it seems to be important the next notions.

Definition 7.1. The equation g(a1, ...,ai1,x,ai+1,an) = ai is with continuous divi- sion on G if for any bG for which g(a1, ...,ai−1,b,ai+1,an) = ai and any open set Ub there exist the open sets U1a1, U2a2,...,Unansuch that for all c1U1, c2U2,..., cnUnthere exits cV such that g(c1, ...,ci1,c,ci+1,cn)=ci.

There exists equations with continuous division and without primitive continuous division.

Example 7.1. Let G = {(x,y) : x2+y2 = 1}and(x,y)·(u,v) = (xy−yv,xv+yu).

Then(G,·)is a compact group. We put zw=z·w·w for any z,wG. Then(G,◦) is a topological groupoid. Consider on G the equations ax=b and ya=b. The equation ya=b has a primitive continuous h(a,b)=b·a1·a1=ba1, where a1 is the inverse element of a in the group(G,·). If u = (cos(φ),sin(φ)) ∈ G and 0 ≤ φ ≤ 2π, then r(u) = (cos(φ/2),sin(φ/2)). In this case λ(a,b) = r(a1·b) is a primitive solution of the equation ax= b. But for the equation ax= b does not exist some continuous primitive solution. The equation ax= b is with continuous solution. The equation ax=b has two distinct solutions for any pair(a,b).

Definition 7.2. A pair(F(X,E, φ), θX)is a topological free E-algebra of a space X in the class V(E, φ)if the following conditions hold:

1. F(X,E, φ)∈V(E, φ)andθX :XF(X,E, φ)is a continuous mapping.

2. IfθX(X)⊆GF(X,E, φ)and GV(E, φ), then F(X,E, φ)=G.

3. For any continuous mapping g : XGV(E, φ)there exists a continuous homomorphism g:F(X,E, φ)G such that g=g◦θX.

Theorem 7.1. For any non-empty space X the free object(F(X,E, φ), θX)exists and is unique.

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