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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2003). pp. 14–35
c© Journal “Algebra and Discrete Mathematics”
On power structures
Ivica Bosˇnjak and Roza´lia Madara´sz
Communicated by Usenko V.M.
Abstract. In general, power structure of a structure A (with
universe A) is an appropriate structure defined on the power set
P(A). There are lot of papers concerning this topic in which power
structures appear explicitly or implicitly. The aim of this paper
is to give an overview of the results that are interesting from the
universal-algebraic point of view.
1. Introduction
There are three basic types of power constructions in universal algebra.
The definitions and a short summary of the most important results are
given in sections 2-4.
In the first type of power construction we ”lift” an operation f on a
set A to the operation f+ on the power set P(A) (see section 2). This
construction is the natural generalization of the multiplication of cosets
in group theory. Some questions related to this construction are studied
in papers: [7], [24], [28], [31], [34] - [36], [41], [45], [48], [55], [60] - [61],
[68], [72]-[75], [77], [79].
A more general construction, i.e. the powering of relational structures
(see section 3), was introduced by Jo´nsson and Tarski in [44]. This con-
struction has proved to be very useful in various areas of algebraic logic
and the theory of non-classical logics. A very comprehensive overview
concerning this type of power constructions can be found in [32].
There are several different ways to lift a relation from a base set to
its power set and they are considered in section 4. A general definition
2001 Mathematics Subject Classification: 08A99.
Key words and phrases: power construction, power algebra, good quotient
relation.
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.I. Bosˇnjak, R. Madara´sz 15
of n-ary power relation was given by Whitney ([82], see also [37]). This
type of construction is widely used in theoretical computer science, in
the context of powerdomains (for an overview see [52] and [67]).
A successful attempt to give a general view on power structures and
to present common ideas used by different authors was made by Chris
Brink in [10]. In [11] he puts power structures in a universal-algebraic
context, thus giving rise to a number of interesting questions and results.
Some of the questions were answered in [3]-[6], [12], [33], [49]-[50]. The
main results from these papers are presented in sections 5-7.
2. Power algebra of an algebra
Let A = 〈A, ·〉 be a groupoid and X,Y ⊆ A. The most natural way to
extend the operation · from A to subsets of A is the following:
X · Y = {x · y | x ∈ X, y ∈ Y }.
This leads us to the general definition of a power operation and a power
algebra:
Definition 1. Let f : An → A. We define f+ : P(A)n → P(A) in the
following way:
f+(X1, . . . , Xn) = {f(x1, . . . , xn) | x1 ∈ X1, . . . , xn ∈ Xn}.
If A = 〈A, {f | f ∈ F}〉 is an algebra, the power algebra P(A) is
defined as:
P(A) = 〈P(A), {f+ | f ∈ F}〉.
Sometimes, it is convenient to exclude the empty set from the universe
of a power algebra. The power algebra defined in this way is denoted by
P+(A).
The first appearance of this concept was probably in the context of
group theory. Namely, if G is a group and N is a normal subgroup of
G, the multiplication of cosets in the factor group G/N is defined as:
xN · yN = xyN for every x, y ∈ G. It is easy to see that this is only a
special case of Definition 1. As any subset of a group was referred to as a
complex, power algebras are sometimes called complex algebras. Also,
some authors (usually in semigroup theory) use the term global of an
algebra. Beside group theory and semigroup theory, power operations
are implicitly used in some other fields. For instance, the set of ideals
of a distributive lattice L again forms a lattice, and meets and joins in
the new lattice are precisely the power operations of meets and joins in
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.16 On power structures
L. In the formal language theory the product of two languages is simply
the power operation of concatenation of words.
The first question that naturally arises here is what properties of a
structure remain invariant under power construction? For example, if an
algebra A satisfies some identity, under what conditions the correspond-
ing power algebra will satisfy the same identity? The complete answer
to this question was given by Gautam (1957).
Theorem 1. (Gautam [31]) Let t1, t2 be different terms. The following
conditions are equivalent.
(1) Each individual variable in the identity t1 ≈ t2 occurs precisely
once on each side of the identity.
(2) For every algebra A, if the identity t1 ≈ t2 is valid in A, it remains
valid in P(A).
Of course, one can ask a more general question: if V is a variety of
algebras, what identities hold in the variety generated by {P(A) | A ∈
V }? It is convenient to introduce the following notation. If V is a variety
of algebras, by P(V ) we denote the variety generated by {P(A) | A ∈ V }.
Similarly, we denote by P+(V ) the variety generated by power algebras
whose universe does not include the empty set. We say that a variety
V is P-closed (P+-closed) if V = P(V ) (V = P+(V )). An identity is
said to be linear if each variable occurs at most once on each side of the
identity. An identity is regular if the same set of variables occurs on
both side of the identity. Gautam’s theorem could be now formulated in
the following way: A variety defined by a single identity is P-closed if
and only if this identity is linear and regular. In general case, we have
the following theorem.
Theorem 2. (Gra¨tzer, Lakser, [36]) Let V be a variety of algebras.
(a) The identities satisfied by P+(V ) are precisely those identities re-
sulting through identification of variables from the linear identities
true in V .
(b) The identities satisfied by P(V ) are precisely those regular iden-
tities resulting through identification of variables from the linear
identities true in V .
There is another subject that has attracted attention of algebraists,
especially semigroup theorists. If K is a class of algebras, and A and
B are isomorphic algebras from K, then P(A) and P(B) are obviously
isomorphic. It is natural to ask whether the converse is true, i.e. is it
true that for any A,B from K it holds:
P(A) ∼= P(B) ⇒ A ∼= B ?
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If the class K has this property, we say that K is a globally deter-
mined class. The first result in this direction comes from 1967, when
T. Tamura and J. Shafer proved that the class of groups is globally de-
termined. It follows immediately from this result that rings are globally
determined. Later, Tamura proved that some other important classes of
semigroups, such as completely simple semigroups, completely 0-simple
semigroups, left (right) zero semigroups and rectangular bands are also
globally determined. Nevertheless, the class of all semigroups is not glob-
ally determined, which was proved by E. M. Mogiljanskaja. In particular,
involutive semigroups are not globally determined ([24]). An important
result was obtained by Y. Kobayashi ([45]). He proved that semilattices
are globally determined, which implies the same result for lattices and
Boolean algebras. Unary algebras have been also investigated in this
context. A. Drapal ([28]) proved that finite partial monounary algebras
are globally determined, while the class of all monounary algebras is not
globally determined. There are also classes of finite algebras which are
not globally determined. Namely, E. Luka´cs ([48]) proved that for every
finite group G, the class of finite G-algebras is globally determined if and
only if G is a cyclic group.
3. Power (complex) algebras of relational structures
A more general construction, i.e. the powering of relational structures,
was introduced by Jo´nsson and Tarski ([44]). They applied this construc-
tion to extend the well-known Stone representation theorem for Boolean
algebras to normal Boolean algebras with operators. In short, a Boolean
algebra with operators is a Boolean algebra with some additional opera-
tions which are additive (distributive on all its arguments, with respect
to Boolean addition). Normality means that the result of the operation
is 0 whenever at least one of the arguments is 0. Above mentioned repre-
sentation theorem of Jo´nsson and Tarski says that every normal Boolean
algebra with operators is isomorphic to a subalgebra of the full power
(complex) algebra of some relational structure. To obtain this result, it
was necessary to incorporate the usual set-theoretic operations into the
definition of power algebra of a relational structure.
Definition 2. If R ⊆ An+1 then the power operation R↑ : P(A)n →
P(A) is defined in the following way:
R↑(X1, . . . , Xn) = {y ∈ A | (∀i ≤ n)(∃xi ∈ Xi) (x1, . . . , xn, y) ∈ R}.
If A = 〈A,R〉 is a relational structure, then the full complex algebra
A+ is defined as A+ = 〈P(A),∪,∩,− , {R↑ | R ∈ R}〉. Every subalgebra
of a full complex algebra will be called a complex algebra over A.
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.18 On power structures
This construction is more general than the first one (see Definition
1) in the following sense: if f is an n-ary operation on A and Gf ⊆ An+1
is the so-called graph of an operation f , i.e.
Gf = {(a1, . . . , an, f(a1, . . . , an)) | ai ∈ A, 1 ≤ i ≤ n},
then G↑f = f+.
This construction has proved to be very useful in various areas of
algebraic logic. Let us mention an example. Relation algebras are
Boolean algebras with three additional operators (a binary, a unary and
a constant) which were equationally defined by Tarski in 1941. They
arose by abstraction from concrete algebras of binary relations (see for
example [43], or [51]). Namely, the set of all binary relations on a set X is
a Boolean set algebra with natural (non-Boolean) operations of relational
composition R ◦S, inversion R−1 and the diagonal relation ∆ = {(x, x) |
x ∈ X}. The class RRA of so-called representable relation algebras
consists of those algebras which are embeddable in a product of algebras
of binary relations as above. Tarski thought in the beginning that the five
defining axioms of relation algebras that he proposed, exactly determine
the class RRA. In fact, RRA is a variety, but it turned out that it is not
finitely axiomatizable. The Jo´nsson-Tarski theory represents the class of
relation algebras RA as complex algebras of an elementary class of some
relational structures, so-called poly-groups. In 1964 R. Lyndon obtained
the first example of a non-representable relation algebra as the complex
algebra of a poly-group (although he did not used that terminology).
Later Jo´nsson constructed non-representable relation algebras using non-
Arguesian projective plane. Developing his ideas, Lyndon associated
a relation algebra to an arbitrary projective plane in which each line
contains at least 4 points and showed that under certain conditions this
relation algebra is not representable. Using similar methods, Maddux
(1978) constructed a relation algebra associated to an arbitrary modular
lattice with 0. Both Lyndon’s and Maddux’s algebras can be viewed as
some complex algebras.
Another discipline where this construction appears is the theory of
non-classical logics. The set theoretic semantics for any well-behaved
logic is based on certain relation structures called Kripke frames. On
the other side, such a logic is characterized by its so-called Lindenbaum-
Tarski algebra and the corresponding variety. The power construction
discussed in this section plays a key role in connecting the logic, its
Kripke semantics and its algebraic semantics. Jo´nsson’s and Tarski’s
work on Boolean algebras with operators and complex algebras came to
the attention of logicians during the early seventies. Before that time
these theories had developed almost independently.
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A very comprehensive overview concerning this type of power con-
structions can be found in [32]. In this paper attention is focused on
varieties whose members can be represented as complex algebras of rela-
tional structures. A recent paper on this topic, [40], concerns the problem
of axiomatization of the class {A+ | A ∈ V }, for a given variety V . For
that purpose the authors use some ”two-player games” and translate the
existence of a winning strategy for one player into a set of first-order
axioms.
4. Power relation of a relation
There are several approaches to the problem of lifting a relation from
a base set to its power set. The early papers mostly deal with special
(binary) relations, such as set-theoretical inclusion or partial orderings
(see for example [8], [9], [81]). For instance, in theoretical computer
science authors use three different ways to construct a power relation of
a partial order. A general definition of n-ary power relations was given
by Whitney in [82], see also [37].
Definition 3. (a) If R ⊆ An, we define R↑k : P(A)n−1 → P(A) as
R↑k(X1, . . . , Xk−1, Xk+1, . . . , Xn) =
= {xk | (∃x1 ∈ X1) . . . (∃xk−1 ∈ Xk−1)
(∃xk+1 ∈ Xk+1) . . . (∃xn ∈ Xn)
(x1, . . . , xk−1, xk, xk+1, . . . , xn) ∈ R},
for all X1, . . . , Xk−1, Xk+1, . . . , Xn ⊆ A.
(b) If R ⊆ An, we define R+ ⊆ P(A)n in the following way:
(X1, X2, . . . , Xn) ∈ R+if
X1 ⊆ R↑1(X2, X3, . . . , Xn)
X2 ⊆ R↑2(X1, X3, . . . , Xn)
...
Xn ⊆ R↑n(X1, X2, . . . , Xn−1)
for all X1, . . . , Xn ⊆ A.
(c) Let A = 〈A,R〉 be a relational structure. The corresponding power
structure is P(A) = 〈P(A), {R+ | R ∈ R}〉.
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.20 On power structures
In particular, if R is a binary relation on A then R+ is a binary
relation on P(A) such that
XR+Y ⇐⇒ (∀x ∈ X)(∃y ∈ Y ) xRy & (∀y ∈ Y )(∃x ∈ X) xRy.
In the binary case this definition is a generalization of Definition 1.
Namely, if f : A → A and Gf = {(a, f(a)) | a ∈ A} is the graph of
f , then G+f = Gf+ . For operations of arity greater than 1, if we exclude
the empty set from the universe of a power algebra, it holds Gf+ ⊆ G+f .
The basic properties of this power construction with respect to the
usual operations on a set of binary relations are given in the following
theorem (see [11]).
Theorem 3. Let R,S ⊆ A2. Then it holds:
(a) R ⊆ S ⇒ R+ ⊆ S+;
(b) (R ∩ S)+ ⊆ R+ ∩ S+;
(c) R+ ∪ S+ ⊆ (R ∪ S)+;
(d) (R ◦ S)+ = R+ ◦ S+;
(e) (R+)−1 = (R−1)+.
Some important properties of relations remain invariant under the
ascent to power relations (reflexivity, transitivity, symmetry), but some
do not. This raises the general question of characterizing those first-order
properties which do remain invariant. The paper of Gra¨tzer and Whit-
ney [37] provides the first result in this direction. This paper concerns
structures in general, where a structure means a set endowed by some
relations and operations (of possibly infinite arity). A variety of struc-
tures is a class axiomatized by a set of atomic formulas in the relevant
first-order language.
Theorem 4. (Gra¨tzer-Whitney [37]) A variety of structures is closed
under power structures iff it is axiomatizable by regular linear atomic
formulas.
This theorem does not solve the general problem stated above, which
still remains open.
In [10] there are a number of results concerning the connection be-
tween power relations, power algebras and quotient algebras, but we will
pay attention to this subject later in section 5.
Power graphs are another topic that has been studied in this con-
text. Namely, any graph can be considered as a relational structure with
one binary relation. Thus a power graph of a graph is simply the power
structure defined according to Definition 3. The notion of a power graph
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is explicitly defined in [1]. In this paper the authors give some results
about automorphisms of power graphs and formulate the problem of
global determinism of graphs. Analogously to the case of algebras (sec-
tion 2), a class K of graphs is globally determined if for every G1, G2 ∈ K
it holds:
P(G1) ∼= P(G2) ⇒ G1 ∼= G2.
It is known that the class of (all) graphs is not globally determined.
It follows from the result of Drapal ([28]) about monounary algebras,
mentioned in section 2. I. Bosˇnjak in his Ph.D. th sis [6] obtained some
results on global determinism of classes of finite graphs. He proved that
the classes of finite trees, finite tournaments and some classes of graphs
with loops are globally determined. Many questions are yet to be an-
swered here. For instance, we do not know whether the class of all finite
graphs is globally determined. Bosˇnjak also made an attempt to connect
the problem of global determinism of graphs with the same problem for
algebras. Namely, there are some well-known constructions that asso-
ciate a special algebra to every graph from a certain class. For exam-
ple, to every tournament we can associate a commutative, idempotent
groupoid. Such groupoids are studied in different contexts (see [16], [22]
[23], [42]). It is proved in [6] that this class of groupoids is globally
determined.
Power graphs have been recently utilized in theoretical computer sci-
ence. In [46] and [47] power graphs are treated as models of concurrent
systems and compared with some other models.
Theoretical computer science gave raise to the study of two alterna-
tive notions of power relation (for an overview see [52] and [67]; see also
[14] and [83]). The first one is associated with so called Hoare pow-
erdomain and the second one with so called Smyth powerdomain.
Generalizations of these two power relations are given in the following
definition.
Definition 4. ([10], [11]) Let X,Y ⊆ A2. We define power relations
R→ and R← in the following way:
XR→Y ⇐⇒ (∀x ∈ X)(∃y ∈ Y ) xRy;
XR←Y ⇐⇒ (∀y ∈ Y )(∃x ∈ X) xRy.
Note that R+ = R→ ∩R←.
These three power relations are cornerstones of the theory of pow-
erdomains. Domains are certain countably-algebraic complete partial
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.22 On power structures
orders that play an important role when assigning a denotation to a de-
terministic program. Namely, if we have a deterministic program, the
denotation of this program is taken to be a continuous mapping D → D,
where (D,⊑) is a countably-algebraic complete partial order. Then the
meaning of a non-deterministic program can be viewed as a continuous
mapping D → P(D). So it is natural to try to turn P(D) into a domain
and for this purpose a power order is used. If ⊑ is the base order on D,
then ⊑→ is associated with the so called Hoare powerdomain, ⊑← with
Smyth powerdomain and ⊑+ with Plotkin powerdomain.
5. Power algebras and quotient algebras
Power algebras are closely related to quotient algebras. Namely, the base
set of the quotient algebra A/θ is included in the base set of the power
algebra P(A). But A/θ is not always a subalgebra of P(A). Also, it
is natural to ask what is the relation between the power algebra P(A)
and the power algebra of a quotient algebra P(A/θ). These and similar
questions are investigated in [11], and in this section, we present some
results from that paper.
Definition 5. ([11]) Let A = 〈A,F 〉 be an algebra, R ⊆ A2, and f is
an n-ary operation from F . We say that R preserves the arguments
of operation f if for all z, y1, . . . , yn ∈ A it holds: If zRf(y1, . . . , yn),
then there exist x1, . . . , xn from A such that xiRyi for all i ∈ {1, . . . , n}
and z = f(x1, . . . , xn). If R preserves the arguments of all the operations
from F and is compatible with all the operations from F , then we say
that R preserves the structure of algebra A.
Theorem 5. ([11]) Let A = 〈A,F 〉 be an algebra and θ ∈ ConA. Then
A/θ is a subalgebra of P(A) if and only if θ preserves the arguments of
all the operations from F .
As we have already mentioned, by lifting an equivalence relation to
the power set, we obtain also an equivalence relation. Moreover, the
following theorem holds:
Theorem 6. ([11]) Let A = 〈A,F 〉 be an algebra and θ ∈ ConA. Then
θ+ ∈ ConP(A).
It is interesting that powering (of algebras and relations) in some
sense commutes with forming quotient algebras. More precisely:
Theorem 7. ([11]) Let A = 〈A,F 〉 be an algebra and θ ∈ ConA. Then
P(A/θ) ∼= P(A)/θ+.
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The following three theorems are ”power versions” of well known
isomorphism theorems.
Theorem 8. ([11]) Let A and B be algebras and ϕ : A → B is an
epimorphism. Then it holds
P(A)/(kerϕ)+ ∼= P(B).
If B is a subalgebra of an algebra A, then [B]θ is defined as [B]θ =
⋃{b/θ | b ∈ B}. The subalgebra of A with the base set [B]θ is denoted
by [B]θ.
Theorem 9. ([11]) Let B be a subalgebra of A, θ ∈ ConA, θ1 = θ
⋂B2,
θ2 = θ
⋂([B]θ)2. Then
P(B)/θ+1 ∼= [P(B)]θ
+/θ+2 .
Theorem 10. ([11]) Let ψ and θ be congruence relations on algebra A
and ψ ⊆ θ. Then
P(A/ψ)
/
(θ/ψ)+ ∼= P(A)/θ+.
6. Good quotient relations
In [11], the author made an attempt to generalize the results quoted in
the previous section to relations on an algebra which are not necessarily
congruence relations. This problem is also studied in [3], [4], [5], [12],
[49], [50], [33].
It is well known that the notion of a quotient algebra A/R is defined
for any universal algebra A = 〈A,F 〉 and any congruence R on A in the
following way:
A/R = 〈A/R, {⌈f ⌉ | f ∈ F}〉,
where A/R is the corresponding quotient set (i.e. the set of all equivalence
classes a/R = {b ∈ A | bRa}, a ∈ A), and for any f ∈ F of arity n,
operation ⌈f ⌉ : (A/R)n → A/R is defined by
⌈f ⌉(a1/R, . . . , an/R) = f(a1, . . . , an)/R. (1)
Of course, if R ⊆ A2 is not a congruence on A = 〈A,F 〉, then opera-
tions ⌈f ⌉ (f ∈ F ) are not necessarily well defined by (1).
Definition 6. Let R be an arbitrary binary relation on A.
(a) For any a ∈ A we define a/R = {b | bRa}. The corresponding
generalized quotient set is A/R = {a/R | a ∈ A}.
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.24 On power structures
(b) Relation ε(R) ⊆ A2 is defined by:
(a, b) ∈ ε(R) ⇐⇒ a/R = b/R.
It is easy to see that definition of ⌈f ⌉ in (1) is ”good” (for every
f ∈ F ) if and only if ε(R) is a congruence on A.
Definition 7. ([33], [49]) Let A = 〈A,F 〉 be an algebra and R ⊆ A2.
(a) We call R a good (quotient) relation on A if ε(R) is a con-
gruence on A. We denote the set of all good relations on A by
G(A).
(b) If R is a good quotient relation on A, the corresponding general-
ized quotient algebra A/R is
A/R = 〈A/R, {⌈f ⌉ | f ∈ F}〉,
where operations ⌈f ⌉ (f ∈ F ) are defined by (1).
Examples
• Any congruence on an algebra is good. In fact, an equivalence
relation is good if and only if it is a congruence.
• Every partial order on A is a good relation on every algebra with
the base set A. Thus every non-trivial algebra has good relations
which are not congruences.
• Every compatible quasi-order (i.e. a reflexive and transitive rela-
tion which is compatible) is a good relation.
• Every structure-preserving relation on an algebra (see Definition
5) is a good relation.
• If A is a set with more than 2 elements, then for any n ≥ 1, there
is a relation R ⊆ A2 and an operation f : An → A such that R is
neither reflexive, nor symmetric, nor transitive nor compatible on
A = 〈A, f〉, but R is a good quotient relation on A.
In order to find out when the partially ordered set G(A) = 〈G(A),⊆〉
is a lattice, we describe the good relations in the following way:
Definition 8. Let R ⊆ A2 and ϕ : A/R → P(A). The relation Rϕ ⊆ A2
is defined as
aRϕb ⇐⇒ a ∈ ϕ(b/R).
Lemma 1. Let A be an algebra and R ⊆ A2. Then R is a good quotient
relation if and only if there is a congruence θ on A and an injection
ϕ : A/θ → P(A) such that R = θϕ.
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Theorem 11. ([4]) Let A be an algebra. The following conditions are
equivalent:
(1) G(A) is a lattice;
(2) EqA = ConA.
(3) G(A) = P(A2).
It is still an open problem to find necessary and sufficient conditions
for an ordered set 〈S,≤〉 to be isomorphic to G(A), for some algebra A.
It can be easily proved that the following ”extended” homomorphism
theorem holds for good quotient relations.
Theorem 12. Let A and B be algebras of the same type. Then B is a
homomorphic image of A if and only if there is a good relation R on A
such that B ∼= A/R.
Not surprisingly, the well known isomorphism theorems cannot be
generalized to the whole class of good relations. So, it is natural to
search for smaller classes of good relations for which some generalized
versions of the isomorphism theorems could be proved. The following
types of good relations proved to be suitable.
Definition 9. Let A be an algebra and R ⊆ A2.
(a) ([12]) We call R a quasi-equivalence on A if for all x,y ∈ A
x/R = y/R ⇐⇒ xRy & yRx.
We denote the set of all quasi-equivalences on A by QEqA.
(b) ([5]) We call R a quasi-congruence on A if R is a good quasi-
equivalence. The set of all quasi-congruences on A is denoted by
QConA.
(c) ([12]) We call R a B-quasi-congruence on A if R is a compat-
ible quasi-equivalence. The set of all B-quasi-congruences on A is
denoted by BQConA.
(d) ([5]) We call R a two-side quasi-congruence on A if for all
x,y,z ∈ A
xRy & yRx & xRz ⇒ yRz.
The set of all two-side quasi-congruences on A is denoted
by TSQConA.
Examples
• Every reflexive and transitive relation on A is a quasi-equivalence
on A.
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.26 On power structures
• Every reflexive and anti-symmetric relation on A is a two-side
quasi-congruence on arbitrary algebra A with the base set A.
In general, TSQConA and BQConA are proper subsets of QConA,
and they are incomparable with each other.
Definition 10. Let A be an algebra, B a subuniverse of A, R ⊆ A2. We
define BR ⊆ A as
BR = {a ∈ A | (∃b ∈ B) a/R = b/R}.
Lemma 2. Let A be an algebra and B a subalgebra of A. Then
(a) If R ∈ QConA then R⋂B2 ∈ QConB.
(b) If R ∈ G(A) then BR is a subuniverse of A.
If R ∈ G(A) and B is a subalgebra of A, then the subalgebra of A
with the universe BR is denoted by BR.
Theorem 13. ([5]) Let B be a subalgebra of an algebra A and R ∈
QConA. If R1 = R
⋂B2 and R2 = R
⋂(BR)2, then
B/R1 ∼= BR/R2.
Definition 11. Let A be an algebra, R,S ∈ TSQConA, R ⊆ S. Rela-
tion S/R ⊆ A/R×A/R is defined as
(a/R, b/R) ∈ S/R ⇔ (a, b) ∈ S.
Lemma 3. Let A be an algebra, R,S ∈ TSQConA, R ⊆ S. Then
S/R ∈ TSQCon(A/R).
Theorem 14. ([5]) Let A be an algebra, R,S ∈ TSQConA, R ⊆ S.
Then
A/R
/
S/R ∼= A/S.
In [12] it is proved that for any algebra A, the set BQConA is closed
under arbitrary intersections. Hence 〈BQConA,⊆〉 is a complete lattice.
It is easy to see that the same is true for the sets QEqA, QConA and
TSQConA.
Theorem 15. ([5]) Let A be an arbitrary algebra with the carrier set
A. Then 〈QEqA,⊆〉, 〈QConA,⊆〉, 〈BQConA,⊆〉, 〈TSQConA,⊆〉 are
algebraic lattices.
It is proved in [12] that ConA is a sublattice of BQConA. ConA
is also a sublattice of QEqA,QConA and TSQConA, which is a conse-
quence of the following theorem.
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.I. Bosˇnjak, R. Madara´sz 27
Theorem 16. ([5]) Let Σ be a set of quasi-congruences of some algebra
A such that ConA ⊆ Σ and Σ is closed under (finite) intersections. If
〈Σ,⊆〉 is a lattice, then 〈ConA,⊆〉 is a sublattice of 〈Σ,⊆〉.
Corollary 1. ([5]) For any algebra A, 〈ConA,⊆〉 is a sublattice of the
following lattices: 〈QConA,⊆〉, 〈BQConA,⊆〉, 〈TSQConA,⊆〉.
Corollary 2. ([5]) For any algebra A = 〈A,F 〉, 〈ConA,⊆〉 is a sublat-
tice of 〈QEqA,⊆〉.
The two corollaries above covers all the cases when one of the lattices
of quasi-congruences is a sublattice of another. Namely, lattices QConA,
BQConA, TSQConA are not necessarily sublattices of QEqA. Also,
lattices BQConA and TSQConA may not be sublattices of QConA.
There exist algebras A = 〈A,F 〉 such that QEqA = QConA, or
QEqA = BQConA and it is not difficult to describe them.
Theorem 17. ([5]) Let A be an algebra and |A| ≥ 3. The following
conditions are equivalent:
(1) QEqA = QConA,
(2) QEqA = BQConA,
(3) EqvA = ConA.
Another problem that might be interesting is to describe algebras
A such that BQConA = ConA (B-quasi-congruence-trivial algebras).
Of course, one can easily notice that for any non-trivial algebra A,
QConA 6= ConA and TSQConA 6= ConA.
Theorem 18. ([5]) Let A be an algebra which has a Mal’cev term (or a
minority term). Then BQConA = ConA.
Corollary 3. ([5]) Let V be a congruence permutable variety. Then for
any A ∈ V , BQConA = ConA.
There are examples which show that there exist B-quasi-congruence-
trivial algebras that are not congruence permutable. Also, there ex-
ist congruence distributive varieties which are not B-quasi-congruence-
trivial.
The following question remains to be answered: is it true that for
any algebraic lattice L, there is an algebra A such that L ∼= BQConA?
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.28 On power structures
7. Powering of good relations
On the ”first level”, good relations do not give new quotient algebras (see
Theorem 12). But their ”inner structures” can be very different and they
do not behave in the same way when we ”lift” them to power structures.
For some good relations R on A, the power relation R+ is good on P(A),
for some other it is not (similarly for R→ and R←).
Definition 12. ([12]) Let A be an algebra and R ⊆ A2.
(a) R is a very good relation on A if R+ is good on P(A). We
denote the set of all very good relations on A by G+(A).
(b) R is Hoare good (or H-good) on A if R→ is good on P(A). we
denote the set of all Hoare good relations on A by G→(A).
(c) R is Smyth good (or S-good) on A if R← is good on P(A). We
denote the set of all Smyth good relations on A by G←(A).
Every congruence is a very good, H-good and S-good relation. More-
over, the following theorem holds.
Theorem 19. Let A be an algebra and R ∈ EqA. The following condi-
tions are equivalent:
(1) R ∈ ConA;
(2) R ∈ G(A);
(3) R ∈ G+(A);
(4) R ∈ G→(A);
(5) R ∈ G←(A).
Compatible quasi-orders are also examples of very good, H-good and
S-good relations. In fact, the following theorem holds.
Theorem 20. ([3]) Let A be an algebra and R ⊆ A2 a quasi-order on
A. The following conditions are equivalent:
(1) R is compatible on A.
(2) R is Hoare good on A.
(3) R is Smyth good on A.
But there are algebras A and very good partial orders on A which
are not compatible on A.
It is not hard to see that every very good (H-good, S-good) relation
is good on A (but the converse is not true). The following power version
of the homomorphism theorem is proved in [12].
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.I. Bosˇnjak, R. Madara´sz 29
Theorem 21. Let A = 〈A,F 〉 be an algebra and R ⊆ A2. Then
(a) If R ∈ G→(A) then P(A)/R→ is a homomorphic image of P(A/R).
(b) If R ∈ G←(A) then P(A)/R← is a homomorphic image of P(A/R).
(c) If R ∈ G+(A) then P(A)/R+ is a homomorphic image of P(A/R).
It is natural to ask what is the relationship between H-good, S-good
and very good relations. The full answer to this question (and some
related questions) is given in [3].
Theorem 22. Every Hoare good relation on an algebra is Smyth good.
Theorem 23. If a relation is Hoare good and Smyth good on an algebra,
then it is also very good.
Corollary 4. Every Hoare good relation on an algebra is very good.
No other positive result can be proved in this direction because there
exist:
• quasi-congruences which are neither H-good nor S-good nor very
good;
• very good relations which are neither H-good nor S-good nor quasi-
congruences;
• S-good relations which are neither H-good nor very good nor quasi-
congruences;
• very good and S-good relations which are neither quasi-congruences
nor H-good;
• H-good relations which are not quasi-congruences.
• H-good quasi-congruences which are not compatible quasi-orders;
• good relations which are neither quasi-congruences nor H-good nor
S-good nor very good;
• very good quasi-congruences which are neither H-good nor S-good;
• very good and S-good quasi-congruences which are not H-good;
• S-good quasi-congruences which are not very good.
It is not a hard problem to construct the listed counter-examples (some
of them could be found in [3] and [6]).
It is easy to see from the definition that the set of good relations
G(A) is closed under complementation and is not closed under ∩ and ∪.
Also, it can be proved that G→(A) and G←(A) are not closed under ∩
and ∪. It is not a surprising fact that both G+(A) and G→(A) are not
closed under complementation. However, despite these facts, it is proved
in [4] that:
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.30 On power structures
Theorem 24. For any algebra A, the set G←(A) is closed under com-
plementation.
As we have already mentioned, if R is a good relation on A, then
R+, R→ and R← are not necessarily good on the corresponding power
algebras. A similar statement can be proved for very good relations.
Namely, if R is a very good relation on A, then R→ (R←, R+) may not
be very good on P(A). Also, if R is a Smyth good relation on A, then
R→ and R+ are not necessarily very good on P(A).
Not all the answers are negative for Smyth good and Hoare good
relations.
Theorem 25. ([4]) Let R be a Smyth good relation on A. Then R← is
a Smyth good relation on P(A).
Theorem 26. ([4]) Let R be a Hoare good relation on A. Then R→ is
a Hoare good relation on P(A).
8. Concluding remarks
Besides the disciplines that are mentioned earlier in the text (such as
theory of non-classical logics or theoretical computer science) there is
another topic where power constructions implicitly appear. The main
concept here is that of hyperoperation (or multioperation, or polyopera-
tion; the terminology varies from author to author). Given a non-empty
set A, a hyperoperation on A is any mapping f : An → P(A). When
one tries to compose such hyperfunctions, it is necessary to lift them to
the power set, i.e. any hyperoperation f : An → P(A) naturally induces
a function f# : P(A)n → P(A) in the following way:
f#(X1, . . . , Xn) = ∪{f(x1, . . . , xn) | xi ∈ Xi}.
Note that the construction f# is in fact the construction from Defi-
nition 2 of Section 3. Namely, there is a natural correspondence between
n-ary hyperfunctions on a set A and n + 1-ary relations on A given in
the following way:
a) If f : An → P(A) then Rf = {(x1, . . . , xn, y) | y ∈ f(x1, . . . , xn)}.
b) If R ⊆ An+1 then fR : An → P(A) such that
fR(x1, . . . , xn) = {y ∈ A | (x1, . . . , xn, y) ∈ R}
Now, if f : An → P(A), then f# = R↑f .
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.I. Bosˇnjak, R. Madara´sz 31
If A is a non-empty set and H is a family of hyperoperations on A,
the ordered pair 〈A,H〉 is called a hyperalgebra (multialgebra, polyal-
gebra). The study of hyperalgebras started in 1934, when so-called hy-
pergroups were introduced in [53]. Since then more then 400 papers
have appeared (see [18] and [80] for a bibliography; see [19], [20], [21],
[65], [66] for some recent results on hypergroups). Most of the papers on
this topic are devoted to special hyperalgebras (hypergroups, hyperrings,
hyperfields, vector hyperspaces, boolean hyperalgebras). Although there
is a number of papers with universal-algebraic flavour (see for example
[13], [17], [27], [29], [30], [38], [39], [58], [59], [62], [63], [64], [78]), there
is still no coherent theory for general hyperalgebras in the literature.
At the end, let us mention that the theory of power structures has
even some non-mathematical applications. Namely, power algebras are
nothing else than special set-valued logic algebras (see [57], [71]), which
have applications to the design of optical, biological and electronical
circuitry.
Acknowledgement
We would like to thank the referee for a careful reading of the manuscript
and some valuable suggestions.
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Contact information
I. Bosˇnjak Institute of Mathematics, University of Novi
Sad, Trg Dositeja Obradovic´a 4, 21000 Novi
Sad, Yugoslavia
E-Mail: ivb@im.ns.ac.yu
R. Madara´sz Institute of Mathematics, University of Novi
Sad, Trg Dositeja Obradovic´a 4, 21000 Novi
Sad, Yugoslavia
E-Mail: rozi@eunet.yu
Received by the editors: 20.01.2003
and final form in 04.07.2003.