Ratio Mathematica
Vol. 34, 2018, pp. 67–76

ISSN: 1592-7415
eISSN: 2282-8214

Algebraic Spaces and Set Decompositions.

Jan Chvalina∗, Bedřich Smetana†

Received: 13-06-2018. Accepted: 24-06-2018. Published: 30-06-2018

doi:10.23755/rm.v34i0.415

c©Jan Chvalina et al.

Abstract

The contribution is growing up from certain parts of scientific work by pro-
fessor Borůvka in several ways. Main focus is on the decomposition theory,
especially algebraized decompositions of groups. Professor Borůvka in his
excellent and well-known book [3] has developped the decomposition (parti-
tion) theory, where the fundamental role belongs to so called generating de-
compositions. Furthermore, the contribution is also devoted to hypergroups,
to algebraic spaces called also quasi-automata or automata without outputs.
There is attempt to develop more fresh view point on this topic.

Keywords: algebraic space; decomposition; join space;
2010 AMS subject classifications: 20N20, 93A10, 20M35.

∗Brno University of Technology, Brno, Czech Republic. chvalina@feec.vutbr.cz
†University of Defence, Brno, Czech Republic. bedrichsmetana@unob.cz

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Jan Chvalina and Bedřich Smetana

1 Introduction
The present contribution is growing up from certain parts of scientific work by

professor Borůvka in several ways. First of all is the decomposition theory, espe-
cially algebraized decompositions of groups. Professor Borůvka in his excellent
and well-known book [3] has developped the decomposition (partition) theory, in
(and on) sets which is applied to decompositions on groupoids and groups where
the fundamental role belongs to so called generating decompositions. It is to be
noted that a decomposition A in a groupoid (G, ·) is called generating if there
exists, to any two-membered sequence of the elements ā, b̄ ∈ A an element c̄ ∈ A
such that āb̄ ⊂ c̄. With the decomposition A in a groupoid (G, ·) there can be
uniquely associate a groupoid denoted (in the mentioned book) by U and defined
such a way that the carrier set of U is the decomposition A and the multiplica-
tion is defined by ā ◦ b̄ = c̄, where ā, b̄, c̄ ∈ A are such elements (i. e. cosets)
that ā · b̄ ⊂ c̄ in the groupoid (G, ·). A special and important case of generating
decompositions on a group (G, ·) created by left on right cosets of an invariant
(normal) subgroup (H, ·) of (G, ·) is the carrier of a factor-group G/H which is
a factoroid created by cosets of the form a · H (or which is the same H · a ) for
an invariant subgroup H of G. On the other hand if left or right decompositions
generated by a subgroup H which is not invariant in a noncommutative group G
are algebraized in a similar way as above, we get multivalued binary operations
on these decompositions which determine a structure called a multigroups or a
hypergroup by the latest terminology. This one has been done by Marty in 1934
and since the time these structures were investigated by many mathematicians in
France, Italy, Greece, Roumania, USA, Canada , Czechoslovakia and elsewhere.

2 Preliminaries
A hypergroup in the sense of Marty is a pair (H, ·) where H is a non-empty

set and · : H ×H →P (́H) (the system of all non-empty subsets off H) is an as-
sociative multioperation (called also a hyperoperation) satisfying the reproduction
axiom: a ·H = H = H ·a for any a ∈ H [11, 12].

A commutative hypergroup (H, ·) is called a join hypergroup or a join space
if it satisfies the exchange condition: For any quadruple a,b,c,d ∈ H such that
a/b∩ c/d 6= ∅ (where a/b = {x ∈ H; a ∈ x · b} and similarly for c/d) we have

(a ·d) ∩ (b · c) 6= ∅.

In the last years investigations of hypergroups which are determined by binary re-
lations (i.e. the binary hyperoperation · is derived by a certain standard way from
a given relation on its carrier set) are of certain interests in investigations on this

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Algebraic Spaces and Set Decompositions.

field. The notion of a join space has been introduced by W. Prenowitz and used by
him and afterwards together with J. Jantosciak to built again several branches of
geometry. In the opinion of professor P. Corsini - which is one of present leading
personalities in the hypergroup theory - the presentation and development of ge-
ometry in the context of join spaces is an important moment in the recent history
of mathematics. There are also close connections of the, mentioned structure to
ternary spaces, especially formed by sets endowed by ternary betweenness rela-
tions, here.

It is to be noted that any abelian group is a join space with single-valued op-
erations. A simple example of a non-trivial join space or a join hypergmup can
be constructed from arbitrary (non-extremal) decomposition of a set: Let A be a
decomposition on a non-empty set A. For any pair of elements x,y ∈ A let us de-
fine x ·y = ā∪ b̄, where ā, b̄ ∈ A are blocks of the given decomposition such that
x ∈ ā,y ∈ b̄. Then it is easy to see that (A, ·) is a join hypergroup (a join space)
in which for a pair x,y ∈ A the fraction x/y is either a block of A containing x or
x/y = A whenever x,y belong to the same block of A.

The algebraic theory of automata is widely elaborated classical discipline; the
golden age or which can be designated from the beginning of sixties up to the end
of tne last century. Nevertheless fundamental publications from the earlier time
due to N. Wiener, J. von Neumann, S. Ginsburg, M. A. Arbib, V. M. Gluškov, R.
E. Kálmán, M. O. Rabin, D. Scott, S. Greibach, K. B. Krohn, J. L. Rhodes, E.
F. More and others, have massive influence on the development of the automata
and artificial languages theory. In spite of studies devoted to finite automata also
infinite automata and their generalizations have been of some interests (cf. Fer-
enc Gécseg, István Peák nad others). It is to be noted that various concepts of a
product of automata (the basic of which has been introduced and studied by M. V.
Gluškov in 1961 as an abstract model of electronic cirquits) are treated in a large
collection of studies devoted to this topics. During the years of investigations of
the mentioned thema, there occure various modifications; most of them can be
generalized to the case of multiautomata or to actions of multistructures. Investi-
gations of automata in connection with multistructures yield more new impulses.
It is evident that infinite antomata without outputs called also quasi-automata are
in fact discrete modifications or “algebraic skelets” of dynamical systems. Ob-
jects of investigations of the mentioned theories can be also considered as special
general systems and they are close to the control theory.

The other connection of this contribution to the research of professor Borůvka
consists in investigations of group and semigroup actions on sets which are sub-
stantial parts of the algebraic concept of an automaton, namely if we concentrate
on changes of states rather than outputs which has been used by professor Borůvka
in his two-parted paper [4]. Automata without outputs are termed also algebraic
spaces (according to Dubreil, Dubreil - Jacobin and Borůvka). So, we can use

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Jan Chvalina and Bedřich Smetana

also this terminology. In accordance with [4] we define an algebraic space with
operators as a triad E = (E,G,α), where E 6= ∅ (a state set or a phase set), G
is a monoid the identity e (in a special case G is supposed to be a group) called
also an input or phase monoid and α : G × E → E is an action (called also a
transition function) wich satisfies two conditions:

1. Identity condition α(e,x) = x for any x ∈ E,

2. Condition of mixed associativity α(b, (α(a,x)) = α(ab,x) for any a,b ∈
G, x ∈ E.

An algebraic space E = (E,G,α) is said to be homogenous if G is acting on
the set E transitively, i.e. for any pair of elements x,y ∈ E there exists a ∈ G
such that α(a,x) = y. Usually an algebraic space E is called homogenous if G
is a group transitively acting on E, which we can called strong homogeneous or
shortly s-homogeneous.

3 Algebraic spaces and hypergroups
We assign to every algebraic space E = (E,G,α) a commutative hypergroup

H(E) = (E,•) in this way: For any pair x,y ∈ E we define

x•y = α(G,x) ∪α(G,y),

where α(G,x) = {α(a,x); a ∈ G} is the trajectory of the element x over the
monoid G. Then the hypergroup H(E) is called a state hypergroup of the algebraic
space E. It is clear that on the state set of any algebraic space E = (E,G,α) there
are defined two totally additive closure operations:

S+, S− : P(E) →P(E)

in this way: S+(X) = α(G,X), S−(X) = {x ∈ X; α(a,x) ∈ X for some a ∈
G} if X is a non-empty subset of the set E and S+(∅) = S−(∅) = ∅ (caled a
source and an successor closure operation, respectively).

The above defined transfer can be extended into functorial if we consider suit-
able morphism between hypergroups (where we use mostly homomorphisms and
good homomorphisms ).

By [18] a hypergroup H is said to be cyclic if for some h ∈ H we have
H =

⋃
k∈N

hk and it is called single-power cyclic (more exactly n-singIe-power

cyclic) if there exist h ∈ H, n ∈ N such that H = hn . In this case the element
his called n-generating. From the above definition of a state hypergroup we get:

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Algebraic Spaces and Set Decompositions.

Proposition 1. An algebraic space E is homogeneous if and only if its state
hypergroup H(E) is 2-single-power cyclic and each element x ∈ E is a 2- gen-
erating element of this hypergroup.2

The following theorem gives necessary and sufficient conditions under which
the state hypergroup of an algebraic space is a join hypergroup:

Theorem 1. Let (E,•) be a state hypergroup of an algebraic space (E,G,α).
Then the following conditions are equivalent:

1. (E,•) is a join hypergroup.

2. For any pair (x,y) ∈ E × E such that x • y ⊆ u2 for a suitable element
u ∈ E, there exists an element v ∈ E with the property v2 ⊆ x2 ∩y2.

3. For any pair (x,y) ∈ E×E such that there exists a pair (a,b) ∈ G×G and
an element u ∈ E with α(a,u) = x, α(b,u) = y, we have α(c,x) = α(d,y)
for some pair (c,d) ∈ G×G.

2

On the contrary to the case of algebraic structures with single-valued oper-
ations in the case of hypergroups there are possible various modifications of the
concept of generating decomposition of the carrier set of a hypergroup. It depends
on the various approaches to the congruence concept for hyperstructures. One of
them is the following notion:

Definition. Let (G, ·) be a hypergroupoid (i. e. · : G×G →P(G) is an arbi-
trary mapping) Let G be such a decomposition on the set G that for any quadruple
a,b,c,d ∈ G with the property a,c ∈ ā, b,d ∈ b̄ for some a,b ∈ G we have
(a · b

[
G
)

= (c · d
[
G
)
; here X

[
Ḡ denotes the closure of the set X in the de-

composition G ([3], 2. 3). Then the decomposition G is called generating (on the
hypergroupoid (G, ·)) or h-generating.

Example 1. Let X be a nonempty set, f : X → X be a mapping. For
x,y ∈ X we put

x ·y = {fn(u); u ∈{x,y},n ∈ N0},

where fn is the n-th iteration of the mapping f. Then it is easy to verify that (X, ·)
is a commutative hypergroup in the above considered sense. Then the decompo-
sition Xf corresponding to a KW-equivalence (Kuratowski -Whyburn - equiva-
lence) r on X is defined by x r y iff fm(x) = fn(x) for some pair m,n ∈ N0 (the
set of all non-negative integers) Then the decomposition Xf is generating on the

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Jan Chvalina and Bedřich Smetana

hypergroup (X, ·).

Example 2. By a deformation of one hypergroupoid (G, ·) onto another one
hypergroupoid (H, ·) we mean a good (also called strong) homomorphism f :
(G, ·) → (H, ·), i.e. for any pair x,y ∈ G we have f(x · y) = f(x) ·f(y). Then
the decomposition G of the hypergroupoid (G, ·) corresponding to deformation f
(i.e. elements x,y ∈ G belong to some element ā ∈ G if an only if f(x) = f(y) )
i.e. the decomposition corresponding to f is h-generating.

4 h-genenerating and Levine‘s decompositions
Now we define a hyperoperation on an h-genenerating decomposition G on a

hypergroupoid (G, ·). For arbitrary pair of elements ā, b̄ ∈ G we put

ā · b̄ = (x.y)[G,

where (x,y) ∈ ā× b̄ is an arbitrary pair.
It is easy to prove that then (G, ·) is a hypergroupoid and that the definition is cor-
rect ( it is independent on the choice of elements x,y). The hypergroupoid (G, ·)
is then called a factor - hypergroupoid on (G, ·) or a hyperfactoroid on (G, ·) or a
hyperfactoroid of (G, ·). Moreover we have:

Theorem 2. Let G be an h-generating decomposition on a hypergroup (G, ·).
Then the hyperfactoroid (G, ·) of (G, ·) is a hypergroup. 2

Now consider an algebraic space with operators E = (E,G,α) with a monoid
G of operators. On the system P(E) of all subsets of E, i.e. the power set of E, we
define a decomposition in this way: Denote S(E) = {K ∈P(E); S + K) = K},
i.e. K ∈ S(E) whenever α(G,K) = K. Now suppose P(E) is a decomposition
of P(E) such that sets X,Y,∈ P(E) belong to some element of P(E) if for any
set M ∈ P(E) such that M = E \ K(a complement) for some K ∈ S(E) we
have X ⊆ M if and only if Y ⊆ M. Then the decomposition P(E) is called a
decomposition of the Levine‘s type or a Levine‘s decomposition of the power set
P(E).

Proposition 2. Let E = (E,G,α) be an algebraic space with operators, P(E)
be the Levine‘s decomposition of power set P(E). Then sets X,Y ∈P(E) belong
to the same element of P(E) if and only if x ∈ X implies α(G,x) ∩Y 6= ∅ and
y ∈ Y implies α(G,y) ∩X 6= ∅.

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Algebraic Spaces and Set Decompositions.

Denote by CS(E) = {M; M ⊆ E,E \M ∈ S(E)} and UE(X) = {M; X ⊆
M,M ∈CS(E)} for any X ∈P(E). Then we get:

Theorem 3. Let E = (E,G,α) be an algebraic space with operators. For any
pair of sets A,B ∈ P(E) we define A • B = UE(A) ∪UE(B) ∪{A,B}. Then
(P(E),•) is a commutative extensive join hypergroup and the Levine‘s decompo-
sition P(E) is h-generating on (P(E),•).

Let f : X → Y be a mapping. We denote by f+ : P(X) → P(Y ) its lifing
into power sets, i.e. we define f+(A) = f(A) = {f(a); a ∈ A} for any non-
empty set A ∈P(X) and f+(∅) = ∅ Then we have

Theorem 4. Let Ei = (Ei,Gi,αi), i = 1, 2, be algebraic spaces with oper-
ators, f : E1 → E2. be a mapping preserving CS - systems of spaces Ei, i.e.
X ∈ CS(E1) implies f(X) ∈ CS(E2). Then f+ is a homomorphism of the hy-
pergroup (P(E1),•) into the hypergroup (P(E2),•). If moreover the mapping f
is surjective and reflects CS- systems, ie. Y ∈ CS(E2) implies f−(Y ) ∈ CS(E1)
(where f−(Y ) is the preimage of the set Y ) we have

f+ : (P(E1),•) → (P(E2),•)

is a deformation, i. e. a good homomorphism of hypergroups and determines a
homomorphism f++ of corresponding factor hypergroups

f++ = (P(E1),•) → (P(E2).

Remark. The closure operations S+,S− : P(E) → P(E) determine a quasidis-
crete or Alexandroff discrete topologies on the state set E of the algebraic space
E, thus some of the above constructions can be expressed in terms of the topo-
logical spaces theory with the use of their special morphisms. Language of the
decomposition theory is in certain sense parallel to algebra of equivalence rela-
tions, however the first approach is useful in the context with coverings of spaces
and with non-associative hyperstructures which are determined by the mentioned
coverings of sets.

There are many papers devoted to hyperstructures - hypergroups and some of
their generalizations in connection with automata and multiautomata. We men-
tion at least papers [6,7,8,9,10] and [12, 13, 14, 15, 16, 17] from references of this
contribution. The mentioned papers contain investigation of transposition hyper-
groups and application of these multistructures for the constructing of actions and
multiactions in connection with some other mathematical concepts.

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Jan Chvalina and Bedřich Smetana

5 Conclusion
Considering the class of all quasiautomata (algebraic spaces) with pointed

monoids as input alphabets (i.e. monoids with distinguished elements) we can
construct multiautomata in such a way that input alphabets are centralizers of dis-
tinguished elements within the given monoids. Hyperoperations on mentioned
alphabets are defined by products of elements using powers of distinguished ele-
ments. Then we obtain a class of multiautomata, where the mentioned construc-
tion - described exactly e. g. in paper [10], page 5 - is functorial, which means that
it preserves homomorphisms; more precisely homomorphisms of quasiautomata
(of algebraic spaces with input monoids) turn out into good homomorphisms of
multiautomata. It is to be noted that multiautomata are serving as suitable tools
for modelling of various processes concernig important mathematical objects and
structures.

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Algebraic Spaces and Set Decompositions.

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