Pseudogroups of transformations on discrete topological spaces
Journal of Applied Mathematics and Computational Mechanics
PSEUDOGROUPS OF TRANSFORMATIONS ON DISCRETE TOPOLOGICAL SPACES
Institute of Mathematics, Czestochowa
University of Technology
Abstract. In all the earlier papers pseudogroups of transformations, generalised inverse semigroups and their connections were considered. This paper is a kind of recapitulation of these problems. Considering pseudogroups of transformations on discrete topological spaces is a step in the same direction. In this paper we notice that domains of functions belonging to a pseudogroup of transformations on discrete topological spaces - when we join an empty set to them create not only a topological space but also a σ-body. We also consider pseudogroups on discrete topological spaces with the finite number of elements. The third problem is the influence of topology on relation of partial order.
Keywords: pseudogroups of transformations, generalised inverse semigroups
The notion of a pseudogroup was formed progressively together with the development of differential geometry. The first mathematicians who realized that a classic notion of a group of transformations was not sufficient for purposes of differential geometry were O. Veblen and J.H.C. Whitehead in 1932. Their definition was improved by J.A. Schouten and J. Haantjes in 1937, S. Gołąb in 1939 and C. Ehresmann in 1947. Both Gołąb’s and Ehresmann’s definitions are good enough to be used at present. It was shown in  that axioms of Ehresmann’s definition can be formulated in an equivalent way which simplify proofs. We used this definition in  to show that a group of transformation can be treated as a pseudogroup. Inverse semigroups and other generalizations of the notion of a group are present not only in differential geometry [3, 4]. This theory is still being developed .
2. Main results
Let us recall the following version of Ehresmann’s definition which is used in differential geometry and can be found in .
Definition 1. A pseudogroup of transformations on a topological space is a set of transformations satisfying the following axioms:
1° Each is a homeomorphism of an open set of onto another open set of ;
2° If , then the restriction of to an arbitrary open subset of the domain of is in ;
3° Let = where each is an open set of . A homeomorphism of onto an open set of belongs to if the restriction of to is in for every ;
4° For every open set of , the identity transformation of is in ;
5° If , then ;
6° If is a homeomorphism of onto and is a homeomorphism of onto and if is non-empty, then the homeomorphism of onto is in .
We will also use the following definition introduced in .
Definition 2. A non-empty set of functions, for which domains are arbitrary non-empty sets, will be called a pseudogroup of functions if it satisfies the following conditions:
|: is a function and is a|
and denotes an inverse relation.
It was shown in  that if is a pseudogroup, then (:) is a topological space and is an Ehresmann pseudogroup of transformations on this topological space. On the other hand, if is an Ehresmann pseudogroup of transformations on a topological space , then is a pseudo-group of functions.
We will use the following definition which we can find in 
Definition 3. We will say that is a Schouten-Haantjes pseudogroup if it satisfies the following axioms:
1° If , and is defined, then ,
2° If and is defined, then .
We notice that a Schouten-Haantjes pseudogroup does not satisfy axioms of Definition 1 because the union of domains may not be a domain of a function belonging to it. In this case we can only say that an intersection of a finite numbers of domains will be a domain of a function which belongs to a Schouten-Haantjes pseudogroup.
Definition 4. A generalized inverse semigroup is a partial groupoid satisfying the following axioms:
holds when one of the sides is defined;
2° For every there exists exactly one such that
We will also need the following definitions and denotations for elements of a generalised inverse semigroup which were introduced in . We will write instead of . For every the only one from 2° of Definition 4 will be denoted by and called a generalised inverse element of , will be called a right identity of and a left identity of . It is obvious that will be then a generalised inverse element of , will be a left identity of and a right identity of . If is a right and left identity for all elements of we say that is an identity. We will say that is an idempotent element when . It was shown in  that for an idempotent element , so it means that its generalised inverse element, the right and left identity are all equal to . It was also shown that the following relation
is a partial order in a generalised inverse semigroup. To prove it we used a lemma saying that if are idempotent elements, the operation is commutative.
It was proved in  that we can obtain an inverse semigroup from every generalised inverse semigroup joining an element . Then (, is a semigroup where the operation is defined in the following way:
|when the operation is defined,|
|in the other case.|
We will also use the theorem which was proved in  and says that if is a pseudogroup of transformations on a topological space , then is a generalised inverse semigroup with identity. Of course we can replace a pseudogroup of transformations by a pseudogroup of functions and the theorem will be true. It was shown in  that even a Schouten-Haantjes pseudogroup is a generalised inverse semigroup. As the definition of Schouten-Haantjes is more general, we can also say that a pseudogroup of functions is a generalised inverse semigroup. It was also proved in  that every generalised inverse semigroup is isomorphic to a Schouten-Haantjes pseudogroup.
Now we will formulate the problems. What can we say when we consider pseudogroups on discrete topological spaces? What can we say when we consider pseudogroups on discrete topological space with a finite number of elements? Is there any influence of topology on algebraic structure as it was in the case of antidiscrete topology . We can formulate the following theorems.
Theorem 1. Domains of functions belonging to an Ehresmann pseudogroup on a discrete topological space together with the empty set create a σ-body.
Proof. As it was shown in , the domains with the empty set consist a topology for all pseudogroups of functions but it is not enough. It will be a σ-body because in a case discrete topological space this family consists of all subsets of the given set which is of course enough to be a σ-body.
Theorem 2. We can obtain an Ehresmann pseudogroup on a discrete topological space with the finite number of elements joining to every cycling group the restrictions its elements to all subsets.
Proof. We have to check all axioms of the definition of a pseudogroup which is not difficult. Of course it is better to use Definition 2. We get what we need using axioms of a group.
Theorem 3. If a generalised inverse semigroup is isomorphic with a pseudogroup of transformations on a discrete topological space then for any idempotent exists smaller or equal to it idempotent for which does not exist smaller idempotent.
Proof. These idempotents for which a smaller idempotent does not exist will be of course idempotents isomorphic to transformations on singletons.
It was shown in  that for generalized inverse semigroups isomorphic to a pseudogroup of transformations on a topological space there exists the largest element in the set of idempotent elements. For pseudogroups of transformations on discrete topological spaces, it is also possible to notice the influence of topology on relation of partial order. Of course Theorem 3 is not true for all pseudogroups. For example, it is not true for pseudogroup of transformations defined on the space of real numbers with natural topology.
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