**SELECTED PROBLEMS AND RESULTS** **OF 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, ...}and*E*be the discrete
sum of topological spaces {*E** _{n}* :

*n*∈ N}. We say that

*E*

*is the space of symbols of*

_{n}*n-ary operations on topological*

*E-algebras. A topological universal algebra of*signature

*E*or a topological

*E-algebra is a non-empty topological spaceG*on which there are given the continuous mappings{e

*nG*:

*E*

*n*×

*G*

*→*

^{n}*G*:

*n*∈N}. The mappings

*e*

*nG*form the algebraical structure on

*G.*

Let*G* be a topological *E-algebra,* *n* ∈ Nand *u* ∈ *E** _{n}*. If

*n*= 0, then

*u(G*

^{0}) =

*e*0G({0} ×

*G*

^{0}) is a singleton and

*u*:

*G*

^{0}→

*G*is a mapping. If

*n*≥1, then we consider the

*n-ary operationu*:

*G*

*→*

^{n}*G, whereu(x*

_{1}, ...,

*x*

*)=*

_{n}*e*

*(u,*

_{nG}*x*

_{1}, ...,

*x*

*).*

_{n}The polynomials are constructed in the following way:

-*E*are polynomials;

- if*n* ∈*N,n* ≥ 1,*u* ∈ *E**n*,*p**i* is an*m**i*-ary polynomial, then *p* = *u(p*1, ...,*p**n*) is an
*m-ary polynomial, where*

*m*=*m*_{1}+*m*_{2}+...+*m** _{n}*and

*p(x*1, ...,*x**m*)=*u(p*1(x1, ...,*x**m1*), ...,*p**n*(x*m**n*−1+1, ...,*x**m*)).

Let *n* ≥ *m* ≥ 1, *p* be an *n-ary polynomial and* *q* : {1,2, ...,*n*} → {1, ...,*m*} be a
mapping. Then*v(x*_{1}, ...,*x** _{m}*) =

*p(x*

*,*

_{q(1)}*x*

*, ...,*

_{q(2)}*x*

*) is an*

_{q(n)}*m-ary term. The polyno-*mials are terms too. If

*u*is an

*n-ary term andv*is an

*m-ary term, thenu(x*1, ...,

*x*

*n*)=

*v(y*

_{1}, ...,

*y*

*) is an identity on*

_{m}*E-algebras.*

Denote by|X| the cardinality of the set *X. Any space is considered to be aT*_{−}1-
space.

Let*i*∈ {−1,0,1,2,3,3^{1}_{2}}.

1

A classKof topological*E-algebra is called aT**i*-quasivarietyif:

- any algebra*G*∈Kis a*T** _{i}*-space,

- if*G*∈Kand*B*is a subalgebra of*G, thenB*∈K,

- the topological product of algebras fromKis a topological algebra fromK,
- if (G,T) ∈ K,T^{′} is a*T**i*-topology on*G* and (G,T^{′}) is a topological*E-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 are*T**i*-spaces, then*V*(E,Ω,*i) is aT**i*-variety. Any*T**i*-variety
is a*T** _{i}*-quasivariety.

A class*V*of*E-algebras is non-trivial if*|G| ≥2 for some*G*∈*V.*

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 Problem*DP*is 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 signature*E* = ⊕{E*n* : *n* ∈ *N}* 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 and*Ω*be*
*a set of identities. Is it true that G admits a structure of topological E-algebra for*
*which G*∈*V(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 reals*R*, or the*
*ring*C*of complex numbers.*

*2. If D is associative and non-commutative, then D is the ring of quaternions*H*.*
*3. If D is non-associative, then D is the ring of octonions*D.

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, let*R*be a topological ring with involution*x* → *x*^{∗}. Denote by*A(R*,∗) the
set*R*^{2} =*R*×*R*with the operations:

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

(x,*y)*·(u,*v)*=(xu−*v*^{∗}*y*,*vx*+*yu*^{∗});

(x,*y)*^{∗}=(x^{∗},−y).

Then*A(R,*∗) is a topological ring with the involution and a topological*R-module.*

The mapping *x* → (x,0) is the natural embedding of the ring*R*into *A(R*,∗). As a
rule, the point *x*∈ *R*is identified by the point (x,0)∈*A(R,*∗) and one may consider
that*R*⊆ *A(R*,∗).

If on the fieldRof reals the identical mapping*x*→*x*^{∗}= *x*is 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 H*n*+1 = *A(H**n*,∗) relatively to the multiplication are
not with division for all*n.*

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 space*G*admits structures of topological *E-algebras.*

For this it is suﬃcient to fix some continuous mapping *e**nG* : *E**n* ×*G** ^{n}* →

*G*for any

*n*∈

*N. In particular, the operation*

*xy*=

*x*determines on

*G*the structure of a topological semigroup with a right identity: the element

*e*∈

*G*is a right (respectively, left) identity if

*xe*=

*x*(respectively,

*ex*=

*x) for anyx*∈

*G.*

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*,*y*∈*A.*

*In this case any continuous mapping*φ : *A*×*A* → *A 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 S^{n}*from*
*the*(n+1)-dimensional Euclidean space E^{n}^{+}^{1}*does not exist the structure of a topo-*
*logical groupoid xy with the identity e*∈*S*^{n}*.*

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

Corollary 2.2. *Let n* ≥ 1, B* ^{n}* = {

*x*∈

*E*

*: ∥*

^{n}*x*∥ ≤ 1}

*, and e*∈

*S*

^{n}^{−}

^{1}⊆

*B*

*⊆*

^{n}*E*

^{n}*. The*

*following assertions are equivalent:*

*1. On the sphere S*^{n}^{−}^{1}*there exists the structure xy of a topological groupoid with*
*the identity e*∈*S*^{n}^{−}^{1}*.*

*2. On Euclidean space E*^{n}*there exists the structure xy of a topological groupoid*
*with the identity e*∈*S*^{n−1}*such that S*^{n−1}*and B*^{n}*are subgroupoids.*

*3. On Euclidean space E*^{n}*there exists the structure xy of a topological groupoid*
*with the identity e*∈*S*^{n}^{−}^{1}*such that B** ^{n}*\ {

*xy*:

*x*,

*y*∈

*S*

^{n}^{−}

^{1}},∅

*.*

*4. n*∈ {1,2,4,8}

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

Assume that*x*·*y*is a structure of a topological groupoid on*S** ^{n−1}*with the identity

*e*∈

*S*

^{n}^{−}

^{1}. Let 0 = (0, ...,0) be the neutral element of the Euclidean space

*E*

*. If*

^{n}*x*∈

*E*

*and*

^{n}*x*,0, then there exists a unique point

*p(x)*∈

*S*

*such that*

^{n−1}*p(x)*=

_{∥}

^{1}

_{x}_{∥}

*x.*

The mapping*h* : *E** ^{n}*\ {0} −→

*S*

*is continuous. Now we put*

^{n−1}*x*∗0 = 0∗

*x*= 0 for each

*x*∈

*E*

*and*

^{n}*y*∗

*z*= ∥

*y*∥ · ∥

*z*∥ ·

*h(y)*·

*h(z) for ally*,

*z*∈

*E*

*\ {0}. Then (E*

^{n}*,∗) is a topological groupoid with the identity*

^{n}*e*and (S

^{n}^{−}

^{1},·), (B

*,∗) are subroupoids.*

^{n}Obviously *x*·*y* = *x*∗*y* for *x,y* ∈ *S*^{n}^{−}^{1}. Implication 1 → 2 is proved. Implication
2→3 is obvious.

Assume that*xy*is a structure of a topological groupoid on*E** ^{n}*with the identity

*e*∈

*S*

^{n}^{−}

^{1}and

*B*

*\ {*

^{n}*xy*:

*x*,

*y*∈

*S*

^{n}^{−}

^{1}},∅. We can suppose that 0∈

*B*

*\ {*

^{n}*xy*:

*x*,

*y*∈

*S*

^{n}^{−}

^{1}}. Then

*x*◦

*y*=

*h(xy) is a structure of a topological groupoid onS*

*with the identity*

^{n−1}*e*∈

*S*

^{n}^{−}

^{1}. Implication 3→1 is proved. The proof is complete.

We need some definitions. A topological quasigroup is a non-empty space*G*with
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 space*G*with two binary operations{+,·}

and the identities*x*+*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 identities*p(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-
tities*p(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
homeomorphism*h*: *X*×*X* →*X*×*X*and a point*c* ∈ *X*such that*h(x*×*X)* = *x*×*X*
and *h(x,x)* = (x,*c) for any* *x* ∈ *X* (M. Choban [28]). The mapping *h* is called a
rectification on*X.*

A space*X*is homogeneous if for any two points*a,b*∈*X*there exists a homeomor-
phism*h** _{ab}* :

*X*→

*X*such that

*h*

*(a)=*

_{ab}*b.*

Let{+,·}be a structure of a homogeneous algebra on a space*G,a*,*b*∈*G*and*x*·*y*

=*c* for all *x* ∈*G. ThenP**a*(x) = *a*·*x,* *Q**a*(u) = *a*+*x*are homeomorphisms, *P*^{−}_{a}^{1}=
*Q** _{a}*,

*P*

*(a)=*

_{a}*c*and

*Q*

*(c) =*

_{a}*a. OnG*there exists a structure {+,·} of homogeneous algebra such that

*c*is the a priori given point. The mapping Ψ(x,

*y)*=(

*x,x*·

*y) is a*homeomorphism of

*G*×

*G*onto

*G*×

*G*such thatΨ(

*x*,

*x)*=(x,

*c) and*Ψ({

*x*} ×

*G)*= {x} ×

*G*for any

*x*∈

*G.*

Assume now that*c* ∈ *X* and*h* :*G*×*G* →*G*×*G*is a homeomorphism such that
*h(x*,*x)*=(x,*c) andh(*{*x*} ×*G)* ={*x*} ×*G*for any *x* ∈*G. Let* *p* :*G*×*G* →*G*be the
projection*p(x,y)*=*y. We putp(h(x,y))*=*x*·yand*p(h*^{−}^{1}(x,*y))*=*x+y*for all*x,y*∈*G.*

Then{+,·}is a structure of a homogeneous algebra on a space*G.*

Let now *f**b*(x)= *x*+*b. Sinceh*^{−1}(G×*c)*={(x,*x) :x*∈*G}*and*h*^{−1}(G× {b}) is the
graphic of the mapping *f**b*, then for*b* , *c* we have *f**b*(x) , *x* for any*x* ∈ *G. Thus*
the mapping *f** _{b}*does not contains fixed points for any

*b*,

*c. In particular,G*is 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 *X* ⊆ *Y*. The mapping*r* : *Y* → *X* is a retraction if*r(x)* = *x* for all *x* ∈ *X.*

If *p* : *Y*^{3} → *Y* is a Mal’cev ternary operation on*Y*, then*q(x*,*y*,*z)* = *r(p(x*,*y*,*z)) is*
a ternary Mal’cev operation on*X. Thus a retract of a Mal’cev algebra is a Mal’cev*
algebra. In particular, any*AR-space admits a structure of a Mal’cev algebra.*

Corollary 2.4. *For any cardinal* τ ≥ 1 *the cube I*^{τ} *is a Mal’cev algebra and it*
*does not admit a structure of homogeneous algebra. For* τ*infinite the space I*^{τ} *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 space*X*admits a structure of a Mal’cev algebra, then*X*is 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(X*^{τ})*is countable for any cardinal* τ*and X*^{m}*admits a structure of a homoge-*
*neous algebra for some cardinal m. Is it true that the space X*^{ℵ}^{0} *admits a structure of*
*a homogeneous algebra?*

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

Remark 2.2. *Let X be a space and A be a non-empty set. Fix a point*0 ∈ *X and*
*an element*α ∈ *A. For any x* ∈ *X we put e(x)* = (x_{β} : β ∈ *A)* ∈ *X*^{A}*, where x*_{α} = *x*
*and x*_{β} = 0*for all* β , α. Then h : *X* −→ *X*^{A}*is an embedding. We identify X and*
*h(X)* *and consider that X* = *h(X)* ⊆ *X*^{A}*. Then the mapping r* : *X** ^{A}* −→

*X, where*

*r(x*

_{β}:β∈

*A)*=

*h(x*

_{α}), is a retraction. Thus the following assertions are equivalent:

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

*2. The space X*^{τ}*admits a structure of a Mal’cev algebra for any cardinal number*
τ*.*

*3. The space X*^{τ}*admits a structure of a Mal’cev algebra for some cardinal number*
τ≥1.

**3.** **PRECOMPACT TOPOLOGIES ON ALGEBRAS**

Fix a discrete signature*E*=⊕{*E**n*:*n*∈*N*}. A topological*E-algebraG*is precom-
pact if*G* is a topological *E-algebra of some Hausdor*ﬀcompact *E-algebra. In this*
section any space is considered to be completely regular.

Let*G* be a topological *E-algebra. A pair (B*, φ) is an *a-compactification or an*
almost periodic compactification of*G*if *B*is a compact*E-algebra,*φ :*G* → *B*is a
continuous homomorphism and the setφ(G) is dense in*B.*

If (B, φ) and (H, ψ) are*a-compactifications ofG, then (H, ψ)*≤(B, φ) if there exists
a continuous homomorphism*g* : *B* → *H* such thatψ=*g*◦φ. For any topological
*E-algebra the classAC(G) of alla-compactifications ofG*is a complete lattice. The
maximal*a-compactification (bhG,b**G*) of*G*is called the Bohr-Holm compactification
of*G. The mappingb** _{G}* :

*G*→

*bhG*is an embedding if and only if

*G*is precompact.

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

Let*G*be a topological*E-algebra andG** _{d}*be the algebra

*G*with the discrete topol- ogy. A pair (H, φ) is called an

*ap-extension ofG*if (H, φ) is an

*a-compactification of*

*G*

*d*and (bhG,

*b*

*G*)≤(H, φ). Thus the class

*EP(G) of allap-extensions ofG*is a com- plete lattice with the maximal element (apG,

*a*

*) and minimal element (bhG,*

_{G}*b*

*). If the space*

_{G}*G*is discrete, then

*apG*=

*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 on*X. The*
algebra*Ba(X) of Baire-measurable sets of the spaceX*is theσ-algebra generated by
the class of functionally closed sets{*f*^{−}^{1}(0) : *f* ∈*C(X)*}of the space*X. A function*
*g* : *X* −→ Cis Baire-measurable if*g*^{−}^{1}(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{*f*^{−}^{1}(H) : *f* ∈*C(X)*,*H* ⊆ C}of the space
*X. A functiong* :*X* −→Cis functionally-measurable if*g*^{−}^{1}(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*

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: *f* ∈*C(H)*}. If (H, φ) is an*ap-extension ofG, thenAP*_{(H,φ)}(G)

= {*f* ◦φ : *f* ∈ *C(H)}. Let* *AP(G)*= *AP*(apG,*a**G*)(G) and *APC(G)* = *APC*(bhG,*b**g*)(G).

Then*AP(G) is the Banach algebra of all almost periodic functions onG*and*APC(G)*
is the Banach algebra of all almost periodic continuous functions on*G.*

If*G*is a topological group, then the function *f* ∈ *F(G) is almost periodic if the*
closure of the set{*f** _{a}* :

*a*∈

*G*}, where

*f*

*(x) =*

_{a}*f*(ax) for all

*a*,

*x*∈

*G, in*

*F(G) is a*compact set.

Remark 3.1. *For a subalgebra L*⊆*AP(G)the following assertions are equivalent:*

*AP1.* *L*= *AP*(H,φ)(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 f* ∈*L, then f* ∈*L.*

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

*1.*Ψ(*f*)= *f*|*X and*∥*f*∥=∥Ψ(*f*)∥*.*

*2.*Ψ(*f* +*g)*= Ψ(*f*)+ Ψ(g)*and*Ψ(*f* ·*g)*= Ψ(*f*)·Ψ(g).

*3. If the sequence* {*f**n* ∈ Φ(β*X) :* *n* ∈ *N*} *converges pointwise to the function*
*f* ∈*F(X), then f* ∈Φ(β*X)and the sequence*{Ψ(*f** _{n}*) :

*n*∈

*N*}

*converges pointwise to*Ψ(

*f*).

*4.*Ψ(C(β*X)*=*C(X),*Ψ(B(β*X)*=*B(X)and*Ψ(Φ(β*X)*=Φ(X).

*5. If X is a topological group, then the function f* ∈ *B(βX)is almost periodic on*
β*X if and only if the function*Ψ(*f*)*is almost periodic on X.*

*Proof.* Assertions 1 - 4 were proved in [27]. Really, for any bounded continuous
function *f* ∈ *C(X) there exists a unique continuous function* β*f* on β*X* such that
*f* = β*f|X. Thus for each functionally-measurable setL*of the space *X* there exists
a functionally-measurable set*L*_{β}of the spaceβ*X* such that*L* = *L*_{β}∩*X. For the set*
*L*_{β} and any point *x* ∈ *L*_{β} there exists a *G*_{δ}-subset *E* ofβX such that *x* ∈ *E* ⊆ *L*_{β}.
Hence, since the space*X*is pseudocompact, the set*L*_{β}is unique. Therefore, for each
function*g* ∈ Φ(X) there exists a unique function β*g* ∈ Φ(β*X) such that* *g* = β*g*|*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 *G** _{H}* = φ(G) the algebra

*G*as a topological subalgebra of the compact algebra

*H. Theap-extension (H, φ) is called*

*B-measurable if*

*AP*

*H*,φ ⊆

*B(G). The*

*ap-extension (H*, φ) is called

*ap-pseudocompact if the spaceG*

*H*is pseudocompact.

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

*Proof.* Let β*G* be the Stone- ˇCech compactification of the pseudocompact group.

ThenβGis a topological group and*G*be a dense subgroup ofβG(see [12]). There ex-
ists a continuous homomorphismϕ:*H*−→β*G*such thatϕ(x)=*x*for any*x*∈*G*⊆*H.*

Assume that *f* ∈ (AP(H,φ)(G)∩Φ(G)). Then, by virtue of Theorem 3.1, there exist
*g* ∈Φ(β*G)) andg*1 ∈*C(H) such that* *f* = *g*|*G*and*g(*ϕ(z))=*g*1(z) for each*z*∈ *H. If*
*B*is a closed subset ofC, then, since the function*g*_{1}is continuous, the set*g*^{−1}_{1} (B) is
closed in*H. Since the mapping*ϕis closed, the setϕ(g^{−1}_{1} (B))= *g*^{−1}(B) is closed in
β*G. Hence the functionsg*and *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*|=2^{α}=τ*. Then G admits exactly*2^{τ}*-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 W* ⊇ *T such that the*
*weight w(G,W)*≥2^{w(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 and*W*be a pseudocompact group topology on*G*
such that*T* ⊆*W. Then the Stone- ˇ*Cech compactification*H*of 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 family*A*of totally bounded group topologies with*

|*A*| =2^{2}^{|}^{G}^{|} *and the spaces*(G,*T*),(G,*w)are not homeomorphic for distinct*(T,*W)* ∈
A.

A cardinal numberτis a strong limit cardinal if 2* ^{m}*< τprovided

*m*< τ. By virtue of Theorem 9.11.2 from ([12], p. 672) it follows:

Corollary 3.1. *Let*τ*be a sequential strong limit cardinal. Then no group of cardi-*
*nality*τ*admits a pseudocompact group topology.*

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

Under Martin’s Axiom MA the infinite Abelian group*G*admits a pseudocompact
group topology if and only if*G*admits a countable compact group topology without

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 G*_{δ}*-topology on apG?*

A space*X*is Dieudonn´e complete if it is complete relatively to the maximal uni-
formity. A subset*L*of a space*X*is bounded if any continuous function *f* : *X* → *R*
is bounded on*L. 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-
sets*F*⊆*G*of*apG*for Abelian*G*was established by H. Leptin [71] and I. Glicksberg
[55].

If*H*is a measurable subgroup of the compact group*G*with the Haar measureλ,
then or*H*is open in*G*orλ(H)=0. Letλbe the Haar measure on*apG, whereG*is a
group with some precompact topology. Then orλ(G) =0, or*G*is not measurable in
*apG*andλ(U)=1 for any measurable set*U*of*apG*which contains*G. For example*
λ(G)=0, if|*G*|<2^{ℵ}^{0}, andλ(U)=1 for any measurable set*U*of*apG*which contains
*G, ifG*admits pseudocompact group topologies. Under which conditionsλ(G)=0?

**4.** **PARATOPOLOGICAL AND**

**SEMITOPOLOGICAL ALGEBRAS**

Fix a discrete signature*E* = ⊕{E*n* :*n* ∈*N}*and the subspaces*S* ⊆ *E* and*P* ⊆ *E.*

An*E-algebraG*with the topology*T* is called:

- an*S*-semitopological*E-algebra if the operationu* :*G** ^{n}* →

*G*is separately con- tinuous for all

*n*∈

*N*and

*u*∈

*S*∩

*E*

*n*;

- a*P-paratopologicalE-algebra if the operationu*:*G** ^{n}* →

*G*is continuous for all

*n*∈

*N*and

*u*∈

*P*∩

*E*

*n*;

- 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 a*T** _{i}*-quasivariety of (P,

*S*)-quasitopological

*E-algebra is defined.*

Theorem 4.1. *(M. Choban [28]) Let V be a T*_{i}*-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 mapping*φ*X* :*X*→*F(X*,*V*)*such that:*

*1. The set*φ* _{X}*(X)

*algebraically generates the E-algebra F*⇁(X,

*V).*

*2. For any continuous mapping g* : *X* →*G* ∈ *V there exists a continuous homo-*
*morphism g*:*F(X,V*)→*G such that g*=*g*◦φ_{X}*.*

The pair (F(X,*V*), φ* _{X}*) is called a free (P,

*S*)-quasitopological

*E-algebra of the*space

*X*in the class

*V.*

The algebra*F(X,V) is abstract free if for any mappingg*:*X*→*G*∈*V*there exists
a homomorphism*g*:*F(X*,*V)*→*G*such that*g*=*g*◦φ*X*.

Problem 4.1. *Assume that there exists a space G*∈*V with a proper open subset.*

*a. Under which conditions the mapping*φ* _{X}* :

*X*→

*F(X,V)is an embedding?*

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

For varieties of topological*E-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. If*S* = *P*= {·}then an*S*-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 a*T** _{i}*-quasivariety of semitopological groups and

*V*

*= {*

_{p}*G*∈

*V*:

*G is a*

*paratopological group},V*

*g*={G∈

*V*:

*G is a topological group}, then:*

1.*V** _{g}* ⊆

*V*

*⊆*

_{p}*V;*

2. If*G*∈*V*and*G**d*is the group*G*with the discrete topology, then*G**d* ∈*V**g*;
3. If (F(X,*V)*, φ*X*),(F(X,*V** _{p}*), φ

*pX*) and (F(X,

*V*

*), φ*

_{g}*gX*) are the free objects of a space

*X, then there exist the continuous homomorphisms*ψ

*X*:

*F(X,V)*→

*F(X,V*

*p*) andθ

*X*:

*F(X*,

*V*

*p*)→

*F(X*,

*V*

*g*) such thatφ

*pX*=ψ

*X*◦φ

*X*andφ

*gX*=θ

*X*◦φ

*pX*;

4. For any completely regular space *X* the mappings ψ*X* and θ*X* are continuous
isomorphisms.

Theorem 4.2. *Let i*∈ {−1,0,1,3^{1}_{2}}*, V be a T**i**-quasivariety of semitopological groups*
*and*Z∈*V. Then for any T*_{i}*-space X:*

*1.*φ* _{X}* :

*X*→

*F(X,V)and*φ

*:*

_{pX}*X*→

*F(X,V*

*p*)

*are embeddings.*

*2. The groups F(X*,*V)and F(X*,*V** _{p}*)

*are abstract free in V and V*

_{p}*respectively.*

*Proof.* Consider the following four cases.

Case 1.*i*=3^{1}_{2}.

This case was proved in [28].

Case 2.*i*=1.

On any set*X*consider the cofinite topologyT*c f* ={*X*} ∪ {*X*\*F* :*F is a finite set*}.
Then (X,T*c f*) is a compact *T*1-space. If*G* is a group, then (G,T*c f*) is a semitopo-
logical compact group. We can assume that*X* = φ*X**d*(X) ⊆ *F(X** _{d}*,

*V*) as a set. Fix a

*T*

_{1}-space

*X. The groupF(X*

*,*

_{d}*V*) is the abstract free group of the set

*X*in the class

*V*. Since (F(X

*d*,

*V)*,T

*c f*) ∈

*V, there exists a unique continuous homomorphism*

*g*:

*F(X*,

*V)*−→ (F(X

*,*

_{d}*V*),T

*c f*) such that

*g(*φ

*X*(x)) =

*x*for each

*x*∈

*X. Theng*is an isomorphism and the object

*F(X,V) is abstract free inV*. Obviously, thatφ

*is an embedding for the space (X,T*

_{X}*c f*). Since any

*T*

_{1}-space

*X*for some cardinal numberτ admits an embedding in (F(X

*d*,

*V),*T

*c f*)

^{τ}, the mappingφ

*is an embedding.*

_{X}Case 3.*i*=0.

Let*D*_{ω} be the groupZwith the topology{∅} ∪ {*U** _{n}* = {

*m*∈ Z :

*m*≥

*n*} :

*n*∈ }. Then

*D*

_{ω}∈

*V*

*p*⊆

*V. Leta,b*be two distinct points of a

*T*0-space

*X. Assume thatU*is open in

*X,a*<

*U*and

*b*∈

*U. Then the mappingg*:

*X*→

*D*

_{ω}, where

*g*

^{−}

^{1}(1) =

*U*and

*g*

^{−1}(0)=

*X*\

*U*is continuous. Thusφ

*:*

_{pX}*X*→

*F(X,V*

*p*) is an embedding. The assertion 1 is proved. The proof of the assertion 2 is proved in [36].

Case 4.*i*=−1.

Let*X*be a space. Let*G*0 be the groupZ×Zwith the topology{∅} ∪ {*V**n* = {*m*∈
Z : *m* ≥ *n} ×*Z : *n* ∈ Z}. The space *X*admits an embedding in*G*^{w(X)}_{0} . Thusφ* _{X}* is
an embedding and we can assume that

*X*= φ

*X*(X) ⊆

*F(X*,

*V*). Let

*G*

*be the group*

_{X}*F(X*

*d*,

*V) with the anti-discrete topology*{∅,

*F(X*

*d*,

*V)}*and

*X*=φ

_{X}*(X)⊆*

_{d}*F(X*

*d*,

*V*) as a set. Then the identical mapping

*f*:

*X*−→

*G*

*, where*

_{X}*f*(x) =

*x*for each

*x*∈

*X*is a continuous mapping and there exists a continuous homomorphism

*g*:

*F(X,V*)−→

*G**X* such that *f* = *g*|*X. Sinceg*is an isomorphism, the group*F(X*,*V) is abstract free*
in the class*V.*

The proof is complete.

Theorem 4.3. *Let i* ∈ {1,3^{1}_{2}}*, V be a non-trivial T**i**-quasivariety of semitopological*
*groups and*Z<*V. Then for any T*_{i}*-space X:*

*1.*φ* _{X}* :

*X*→

*F(X,V)and*φ

*:*

_{pX}*X*→

*F(X,V*

*p*)

*are embeddings.*

*2. The groups F(X*,*V)and F(X*,*V** _{p}*)

*are abstract free in V and V*

_{p}*respectively.*

*Proof.* Consider the following two cases.

Case 1.*i*=3^{1}_{2}.

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 group*G* with a topology is called a*left*(respectively,*right) topological group*
if the left translation *L**a*(*x)* = *ax* (respectively, the right translation*R**a*(x) = *xa) is*
continuous for any*a*∈*G.*

A classVof left topological groups is called a*T**i*-quasivariety of left topological
groups if:

(LF1) the classVis multiplicative;

(LF2) if*G*∈Vand*A*is a subgroup of*G, thenA*∈V;
(LF3) every space*G*∈Vis a*T**i*-space;

(LF4) if*G* ∈V,Tis a compact*T**i*-topology on*G*and (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*∈ {−1,0,1,3^{1}_{2}}*, V be a T*_{i}*-quasivariety of left topological groups*
*and*Z∈*V. Then for any T**i**-space X:*

*1.*φ*X* :*X*→*F(X*,*V)and*φ*pX*:*X*→*F(X*,*V** _{p}*)

*are embeddings.*

*2. The groups F(X,V)and F(X,V**p*)*are abstract free in V and V**p**respectively.*

Corollary 4.2. *Let i* ∈ {1,3^{1}_{2}}*, V be a non-trivial T*_{i}*-quasivariety of left topological*
*groups and*Z∈*V. Then for any T**i**-space X:*

*1.*φ*X* :*X*→*F(X*,*V)and*φ*pX*:*X*→*F(X*,*V**p*)*are embeddings.*

*2. The groups F(X*,*V)and F(X*,*V** _{p}*)

*are abstract free in V and V*

_{p}*respectively.*

The following assertion completes Theorem 4.3 and Corollary 4.2.

Lemma 4.1. *Let G be a left topological group and for any x*∈*G there exists n(x)*∈N
*such that x** ^{n(x)}*=

*e. Then G is a T*1

*-space.*

*Proof.* Any finite*T*0-space contains a closed one-point subset. Thus any finite left
topological group is a*T*_{1}-space. By hypothesis, any point*a* ∈*G*is contained in the
finite subgroup*G(a)* ={a* ^{i}* : 0≤

*i*≤

*n(a)}. Thus*{e}is a closed subset of the group

*G*and

*G*is a

*T*

_{1}-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 T*_{0}*-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].*

Let*S* ⊆*E,G*be an*E-algebra,n*≥1, *j*∈ {1,2, ...,*n*},*u*∈*E** _{n}*∩

*S*and

*a*

_{1},

*a*

_{2}, ...,

*a*

*∈*

_{n}*G. We putR(G*,

*j*,

*u*,

*a*1, ...,

*a*

*n*)={

*x*∈

*G*:

*u(a*1, ...,

*a*

*j*−1,

*x*,

*a*

*j*+1, ...,

*a*

*n*)=

*a*

*j*}.

The *E-algebraG*is called an*S*-simple*E-algebra if for alln* ≥ 1, *j*∈ {1,2, ...,*n*},
*u* ∈ *E**n* ∩*S* and *a*1,*a*2, ...,*a**n* ∈ *G* we have *R(G,j,u,a*1, ...,*a**n*) = *G* or the set
*R(G*, *j*,*u*,*a*_{1}, ...,*a** _{n}*) is finite.

All quasigroups are simple algebras.

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

*1. the mapping*φ*X* :*X*→*F(X*,*V*)*is an embedding.*

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

*Proof.* Let*G* ∈ *V. Denote by* T*c f* = {∅} ∪ {*G*\*F* : *F is a finite set*} the co-finite
topology on*G. SinceG*is an*S*-semitopological*S*-simple*E-algebra the operation*
*u*:*G** ^{n}*→

*G*is separately continuous for all

*n*∈

*N*and

*u*∈

*S*∩

*E*

*n*. Thus (G,T

*c f*)∈

*V*. Fix a non-empty

*T*

_{1}-space

*X. Denote byX*

*the set*

_{d}*X*with the discrete topology.

Then the *E-algebra (F(X**d*,*V*), φ_{X}* _{d}*) is the abstract free algebra of the space

*X*in the

class*V*. Let*G** _{X}* be the algebra

*F(X*

*,*

_{d}*V) with the co-finite topology*T

*c f*. Then

*G*

*∈*

_{X}*V, the mapping*

*g*= φ

_{X}*:*

_{d}*X*−→

*G*

*X*is continuous and an injection. There exists a continuous homomorphism

*h*:

*F(X*,

*V*) −→

*G*

*X*such that

*h(*φ

*X*(

*x))*=

*g(x) for each*

*x*∈

*X. Henceg*is an isomorphism and the algebra

*F(X*,

*V) is abstract free. Since*

|X| ≤ |G*X*|, then for some cardinalτthe space *X*admits an embedding in*G*^{τ}* _{X}*. Thus
the mappingφ

*X*:

*X*→

*F(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 T*_{i}*-quasivariety of semitopolog-*
*ical groups. Are Theorems 4.2 and 4.3 true?*

*b. Let i* ∈ {2,3}*and V be a non-trivial T*_{i}*-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 G*_{δ}*-subspace of some*
*pseudocompact space, then G is a topological group and a dense G*_{δ}*-subspace of*
*some pseudocompact space.*

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

A quasigroup*G*with a topology is called:

- a paratopological quasigroup if the multiplicative operation{·} and the transla-
tions*l**a*=*l(a,x),r**a*=*r(x,a),a*∈*G, are continuous;*

- a semitopological quasigroup if the translations *L**a*(x) = *a*· *x,* *R**a*(*x)* = *x* ·*a,*
*l** _{a}*=

*l(a*,

*x),r*

*=*

_{a}*r(x*,

*a),a*∈

*G, 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 *L**a*(*x)* = *a*·*x,*
*R**a*(x) = *x*·*a,* *l**a* = *l(a*,*x),r**a* = *r(x*,*a),a* ∈ *G, are homeomorphisms. Moreover,*
*l** _{a}*=

*L*

^{−}

_{a}^{1}and

*r*

*=*

_{a}*R*

^{−}

_{a}^{1}for each

*a*∈

*G.*

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*−→*G*such that*L**a*,*R**a* ∈*P(G*,·) for each*a*∈*G.*

A*T*-groupoid (or a Toyoda groupoid) is a non-empty set*G*with one binary oper-
ation{·}and four unary operations{a1,*a*2,*b*1,*b*2}such that:

if*x*◦*y*=*a*1(*x)*·*b*1(y)), then (G,◦) is a group;

*a*_{1}(a_{2}(*x))*=*b*_{1}(b_{2}(x))=*x*for each*x*∈*G;*

{*a*_{1},*a*_{2}} ∩*P(G*,·),∅and{*b*_{1},*b*_{2}} ∩*P(G*,·),∅.

In this case we say that (G,◦) is the group associated to the *T*-groupoid
(G,·,*a*_{1},*a*_{2},*b*_{1},*b*_{2}). By definitions,*a*_{2} =*a*^{−}_{1}^{1}and*b*_{2}=*b*^{−}_{1}^{1}.

Any*T*-groupoid is a quasigroup.

Let (G,◦) be the topological group associated to a topological *T*-groupoid
(G,·,*a*1,*a*2,*b*1,*b*2). 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*
{*a*1,*a*2,*b*1,*b*2,*c*1,*c*2}is a topological*T*-groupoid, then:

- we have*x*·*y*=*a*_{2}(x)◦*b*_{2}(y),*l(x*,*y)*=*b*_{1}(a_{2}(x)^{−}^{1}◦*y) andr(x*,*y)*=*a*_{1}(x◦*b*_{2}(y)^{−}^{1});

- there exists many structures of the kind{a1,*a*2,*b*1,*b*2}on*G;*

- all topological groups associated to the*T*-groupoids (G,·,*a*_{1},*a*_{2},*b*_{1},*b*_{2}) are topo-
logically isomorphic.

Therefore any topological*T*-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
topological*T*-groupoid is completely regular provided it is a*T*_{0}-space.

A*T*-groupoid (G,·,*a*1,*a*2,*b*1,*b*2) with a topology is called:

- a topological*T*-groupoid if the operation (G,·,*a*1,*a*2,*b*1,*b*2) are continuous and
*G*is a topological quasigroup;

- a paratopological*T*-groupoid if the operation{·,*a*1,*a*2,*b*1,*b*2}are continuous;

- a semitopological*T*-groupoid if the multiplicative operation{·}is separate con-
tinuous and the operation{a1,*a*2,*b*1,*b*2}are continuous.

If a (G,·,*a*1,*a*2,*b*1,*b*2) is a semitopological *T*-groupoid, then the operations
{*a*_{1},*a*_{2},*b*_{1},*b*_{2}}are homeomorphisms. Moreover, if a *T*-groupoid (G,·,*a*_{1},*a*_{2},*b*_{1},*b*_{2})
is a semitopological quasigroup, then the operation {a1,*a*2,*b*1,*b*2} are homeomor-
phisms.

We mention that a*T*-groupoid (G,·,*a*1,*a*2,*b*1,*b*2) with topology:

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

- is a paratopological*T*-groupoid if and only if (G,·,*r*,*l) (G*,·*r*,*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,*a*2,*b*1,*b*2}is considered a*T*-groupoid
too. Therefore:

- any semitopological group is a a semitopological*T*-groupoid;

- any paratopological group is a a paratopological*T*-groupoid;

- any topological group is a a topological*T*-groupoid.

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

- any medial quasigroup is a*T*-groupoid;

- any semitopological medial quasigroup is a a semitopological*T*-groupoid;

- any paratopological medial quasigroup is a a paratopological*T*-groupoid;

- any topological medial quasigroup is a a topological*T*-groupoid.

Theorem 5.3. *Let*K*be a class of topological spaces. Then:*

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

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

*Proof.* Let*H* = (G,◦) be the associated group at the*T*-groupoid (G,·,*a*1,*a*2,*b*1,*b*2)
with the topology and{*a*_{1},*a*_{2},*b*_{1},*b*_{2}}.

Then:

-*H*is a semitopological group if and only if*G*is a semitopological*T*-groupoid ;
-*H*is a paratopological group if and only if*G*is a paratopological*T*-groupoid;

-*H*is a topological group if and only if*G*is a topological quasigroup, i.e a topo-
logical*T*-groupoid.

The proof is complete.

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

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 space*X* is called τ-solvable if there
exists a family{X_{α} :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 quasigroup*G. The topology*Tis weakly bounded if for
any non-empty set*U* ∈Tthere exists a finite set*L*⊆*G*such that*G* = *L*·*U. We do*
not suppose that (G,T) is a topological, or a semitopological quasigroup.

Example 6.1. *Denote by T*_{1}(G)={*X*} ∪ {*X*\*F* :*F is a finite subset of G*}*the minimal*
*T*1*-topology on the quasigroup G, i.e. the cofinite topology on G. If b* ∈ *G, then*
*T*_{0}(G,*b)*={*U* ∈*T*_{1}(G) :*b*∈*U*}*is a T*_{0}*-topology on G. Then:*

*- if*T⊆*T*1(G), thenT*is a weakly bounded topology on G;*

*-*(G,*T*1(G))*is a semitopological quasigroup;*

*- if the set G is infinite, then*(G,*T*_{1}(G))*is not a paratopological quasigroup;*

*- let G contains two distinct points and b*∈*G, then*(G,*T*0(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 {B_{µ} : µ ∈ *M}* *of subsets of G such*
*that:*

*1.*|*M*|=|*G*|*.*

*2. G*=∪{*B*_{µ} :µ∈*M*}*.*

*3.*(G\*B*µ)·*K* ,*G for all*µ∈*M and every finite subset K of G.*

*4. The sets*{B_{µ} :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 for*IP-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**

Let*E*be a signature. If*n*≥1,*g*∈*E**n*and 1≤*i*≤*n, then*

*g(a*_{1}, ...,*a** _{i−1}*,

*x*,

*a*

*, ...,*

_{i+1}*a*

*)=*

_{n}*a*

*is an equation on*

_{i}*E-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 *a*_{1},*a*_{2}, ...,*a** _{n}* ∈

*G*there exists

*b*∈

*G*such that

*g(a*

_{1}, ...,

*a*

_{i}_{−}

_{1},

*b*,

*a*

_{i}_{+}

_{1},

*a*

*) =*

_{n}*a*

*i*. By

*V(E,uφ) we denote the class of all algebras*

*G*∈

*V(E, φ) on which the*equations

*e(g*,

*n*,

*i)*∈ φ are unique solutions. Let

*e(g*,

*n*,

*i) be an equation from*φ. We say that there exists a primitive solution on

*G*of this equation if there exists a term

*h(y*

_{1},

*y*

_{2}, ...,

*y*

*) such that*

_{n}*g(a*

_{1}, ...,

*a*

*,*

_{i−1}*h(a*

_{1}, ...,

*a*

*, ...,*

_{i}*a*

*),*

_{n}*a*

*, ...,*

_{i+1}*a*

*) =*

_{n}*a*

*for any*

_{i}*a*1,

*a*2, ...,

*a*

*n*∈

*G. LetV*(E, φ,Π) be the class of

*E-algebrasG*∈

*V(E, φ) with*the primitive solutions for all equations fromφ. Obviously

*V(E*, φ,Π)⊆

*V(E*, φ). In some cases we may extend the signature

*E*and consider that the solutions fromφare operations from the signature.

There exists*E-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(a*1, ...,*a**i*−1,*x,a**i*+1,*a**n*) = *a**i* *is with continuous divi-*
*sion on G if for any b* ∈*G for which g(a*_{1}, ...,*a** _{i−1}*,

*b*,

*a*

*,*

_{i+1}*a*

*) =*

_{n}*a*

_{i}*and any open set*

*U*∋

*b there exist the open sets U*1∋

*a*1

*, U*2 ∋

*a*2

*,...,U*

*n*∋

*a*

*n*

*such that for all c*1∈

*U*1

*,*

*c*2 ∈

*U*2

*,..., c*

*n*∈

*U*

*n*

*there exits c*∈

*V such that g(c*1, ...,

*c*

*i*−1,

*c*,

*c*

*i*+1,

*c*

*n*)=

*c*

*i*

*.*

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

Example 7.1. *Let G* = {(*x,y) :* *x*^{2}+*y*^{2} = 1}*and*(x,*y)*·(u,*v)* = (xy−*yv,xv*+*yu).*

*Then*(G,·)*is a compact group. We put z*◦*w*=*z*·*w*·*w for any z*,*w*∈*G. Then*(G,◦)
*is a topological groupoid. Consider on G the equations a*◦*x*=*b and y*◦*a*=*b. The*
*equation y*◦*a*=*b has a primitive continuous h(a,b)*=*b*·*a*^{−}^{1}·*a*^{−}^{1}=*b*◦*a*^{−}^{1}*, where*
*a*^{−}^{1} *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(a*^{−}^{1}·*b)* *is a*
*primitive solution of the equation a*◦*x*= *b. But for the equation a*◦*x*= *b does not*
*exist some continuous primitive solution. The equation a*◦*x*= *b is with continuous*
*solution. The equation a*◦*x*=*b has two distinct solutions for any pair*(a,*b).*

Definition 7.2. *A pair*(F(X,*E*, φ), θ*X*)*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)*and*θ*X* :*X* →*F(X*,*E*, φ)*is a continuous mapping.*

*2. If*θ*X*(X)⊆*G*⊆*F(X,E, φ)and G*∈*V(E, φ), then F(X,E, φ)*=*G.*

*3. For any continuous mapping g* : *X* → *G* ∈ *V(E*, φ)*there exists a continuous*
*homomorphism g*:*F(X,E, φ)*→*G such that g*=*g*◦θ*X**.*

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