Galois and Pataki Connections on Generalized Ordered Sets

In this paper, having in mind Galois and Pataki connections, we establish several basic theorems on increasingly seminormal and semiregular functions between gosets. An ordered pair ( ) ( ) ≤ = ≤ , X X consisting of a set X and a relation ≤ on X is called a goset (generalized ordered set). A function f of one goset X to another Y is called increasingly upper g-seminormal, for some function g of Y to X, if ( ) y x f ≤ implies ( ). y g x ≤ While, the function f is called increasingly upper φ -semiregular, for some function φ of X to itself, if ( ) ( ) v f u f ≤ implies ( ). v u φ ≤ The increasingly lower seminormal (semiregular) functions are defined by the reverse implications. Moreover, a function is called increasingly normal (regular) if it is both increasingly upper and lower seminormal (semiregular). The results obtained extend and supplement several former results of O. Ore and the present author on Galois and Pataki connections. Namely, the pairs ( ) g f , and ( ) φ , f may be called increasing Galois and Pataki connections if the function f is increasingly g-normal and φ -regular, respectively.


Introduction
Ordered sets and Galois connections occur almost everywhere in mathematics [8]. They allow of transposing problems and results from one world of our imagination to another one.
In [28], having in mind a terminology of Birkhoff [1, p. 1], an ordered pair consisting of a set X and a relation ≤ on X is called a goset (generalized ordered set).
In particular, a goset ( ) is called a proset (preordered set) if the relation ≤ is reflexive and transitive. And, a proset is ( ) ≤ X called a poset (partially ordered set) if the relation ≤ is in addition antisymmetric.
In [35], motivated by an ingenious observation of Schmidt [20, p. 209] a function f of one goset X to another Y is called increasingly upper g-seminormal, for some function and . Y y ∈ While, motivated by a fundamental definition of Pataki [17, p. 160], the function f is called increasingly upper ϕ -semiregular, for some function ϕ of X into itself, if Thus, if f is increasingly upper g-seminormal, then by taking , f g = ϕ we can at once see that Therefore, f is increasingly upper ϕ -semiregular.
The increasingly lower seminormal (semiregular) functions are defined by the reverse implications. Moreover, a function is called increasingly normal (regular) if it is both increasingly upper and lower seminormal (semiregular). Now, in particular, the pairs ( ) may also be naturally called increasing Galois and Pataki connections between the gosets X and Y if the function f is increasingly g-normal and ϕ -regular, respectively.
Moreover, we can at once see that a function f of one goset X to another Y is increasingly lower X ∆ -semiregular if and only it is increasing in the usual sense that v u ≤ implies ( ) In this respect, it is also worth mentioning that if f is an increasingly ϕ -regular function of one proset X to another Y, then f is increasing and ϕ is a closure operation (See [33].) Here, a function ϕ of a goset X to itself is called a closure operation if it is increasing, extensive and lower semiidempotent in the sense that ( ) Thus, ϕ is idempotent if X is a poset.
The importance of regular functions is also apparent from the fact that a function ϕ of a proset X to itself is a closure operation if and only if ϕ is increasingly ϕ -regular.
That is, In [33], we have also proved that a function f of one proset X to another Y is increasingly ϕ -regular if and only if ϕ is a closure operation on X such that Several interesting characterizations of increasingly regular and normal functions have also been established in [31,35]. Moreover, some closely related results for increasing functions and closure operations have also been proved in [40]. Now, by improving and extending some of these results, we shall, for instance, prove that for a function f of a sup-complete proset X to an arbitrary one Y, the following assertions are equivalent: (1) f is increasingly normal, (2)  Moreover, it is noteworthy that if X and Y are supposed to be only arbitrary gosets, then (1) already implies the second statement of (6). In [40], for a function f of one goset X to another, we have only prove that f is increasing if and only if At the end of this paper, we shall also offer some generalizations of increasingly seminormal functions on posets to relations on relator spaces of the form ( ) ( ), where X and Y are sets and R is a family of relations on X to Y.

A Few Basic Facts on Relations
then we may simply say that F is a relation on X. In particular, is called the identity relation on X.
If F is a relation on X to Y, then by the above definitions we can also state that F is a relation on . Y X ∪ However, for several purposes, the latter view of the relation F would be quite unnatural.
If F is a relation on X to Y, then for any , : : , Thus, the Axiom of Choice can be briefly expressed by saying that every relation has a selection.
For any relation F on X to Y, we may naturally define two set-valued functions, F of X to ( ) can be identified with relations on X to Y. While, functions of ( ) are usually more powerful tools than relations on X to Y [38,46].
However, they are frequently less flexible.
The latter notation will not cause confusions, since thus we also have .
can be naturally defined such that Thus, the operations c and −1 are compatible in the sense ( ) ( ) .
we also have ( Hence, by taking one can see that the box and composition products are actually equivalent tools. However, the box product can be immediately defined for arbitrary families of relations.
is a tolerance relation on X.
Note that Now, for any relation R on X, we may also naturally define Moreover, we may naturally define Thus, ∞ R is the smallest preorder relation on X containing R [12] .

A Few Basic Facts on Generalized Ordered Sets
According to [28], an ordered pair consisting of a set X and a relation ≤ on X, will be called generalized ordered set, or an ordered set without axioms. And, we shall usually write X in place of ( ).

≤ X
In the sequel, a generalized ordered set ( ) ≤ X will, for instance, be called reflexive if the relation ≤ is reflexive. Moreover, the generalized ordered set be called the dual of ( ). ≤ X Having in mind the terminology of Birkhoff [1, p. 1], a generalized ordered set may be briefly called a goset. Moreover, a preordered (partially ordered) set may be called a proset (poset).
Thus, every set X is a poset with the identity relation .
Moreover, X is a proset with the universal relation . 2 X And every subfamily of the power set ( ) X P of X is a poset with the ordinary set inclusion . ⊆ The usual definitions on posets can be naturally extended to gosets [28]. (And, even to arbitrary relator spaces [27] which include ordered sets [7], context spaces [10], and uniform spaces [9] as the most important particular cases.) For instance, for any subset A of a goset X, we may naturally define In the sequel, by identifying singletons with their elements, we shall, for instance, Therefore, the set-functions lb and Φ form a Pataki connection between the poset ( ) X P and its dual in the sense of [31, Remark 3.8] suggested by a fundamental unifying work of Pataki [17] on the basic refinements of relators studied each separately by the present author in [24] .
By [33], the letter fact implies that , lb lb Φ = and the function Φ is a closure operation on the poset ( ) in the sense of [1, p. 111]. By an observation, attributed to Dedekind by Erné [8, p. 50], this is equivalent to the requirement that the function Φ with itself form a Pataki connection between the poset ( ) X P and itself.

Some Further Results on Gosets
Concerning minima and maxima, and infima and suprema, one can easily prove the following theorems.

Theorem 3.1. For any subset A of a goset X, we have
Remark 3.2. By this theorem, for instance, we may also naturally define is just the family of all maximal elements of A.

Theorem 3.4. For any subset A of a goset X, we have
then X is antisymmetric.
In [29], by using the notation we have first proved that a reflexive goset X is antisymmetric if and only if Quite similarly, a goset X may, for instance, be also naturally called Thus, the set N of all natural numbers is min-, but not inf-complete. While, the Now, as an immediate consequence of Theorem 3.4, we can state the following straightforward extension of [1, Theorem 3, p. 112]. Theorem 3.9. For a goset X, the following assertions are equivalent: (2) X is sup-complete.

Remark 3.10. Similar equivalences of several modified inf-and sup-completeness
properties of gosets have been established in [4] and [3].
Moreover, as a consequence of the corresponding definitions, we can also state Theorem 3.13. For a goset X, the following assertions are equivalent: (1) X is reflexive and linear, Hence, it is clear that in particular we also have Corollary 3.14. If X is a min-complete (max-complete) goset, then X is reflexive and linear.
The importance of reflexive, linear, and antisymmetric gosets is apparent from the following To check (2), note that if , v u ≤ / then by Theorem 3.13 we have . u v ≤ Moreover, by the reflexivity of X, we also have . u v ≠ Therefore, we also have .
Now, as an immediate consequence of this theorem, we can also state the following very particular Galois-type connection.

Corollary 3.16. If X is a reflexive, linear and antisymmetric goset, then for any
can at once see that ( ) Y Y ≤ is also a goset which inherits several basic properties of the original goset ( ).
Moreover, concerning subgosets, we can also easily prove the following also holds, by using (3) we can see that then by using (3) we can see that This shows that Hence, since , Y ∈ α by using (3) we can already infer that Thus, by Theorem 3.3,

Remark 3.19.
In connection with (5), Tamás Glavosits, my PhD student, showed that the corresponding equality need not be true even if X is finite poset.
For this, he took and considered the preorder ≤ on X generated by the relation Thus, he could at once see that and thus ( )

Increasing Functions of One Goset to Another
Increasing functions are usually called isotone, monotone, or order-preserving in algebra. Moreover, in [7, p. 186] even the extensive maps are called increasing. However, we prefer to use a terminology of analysis [21, p. 128].
Definition 4.1. If f is a function of one goset X to another Y, then we say that: Therefore, the study of decreasing functions can be traced back to that of the increasing ones. In [40], by proving the following statements, we have shown that almost the same is true in connection with the strictly increasing ones.

Theorem 4.3.
For a function f of one goset X to another Y, the following assertions hold: (1) f is strictly increasing if f is injective and increasing, (2) f is injective if X is linear and f is strictly increasing. In [40], concerning strictly increasing functions, we have also proved However, it is now more important to note that, by [40], we also have For a function f of one goset X to another Y, the following assertions are equivalent: (1) f is increasing, Note that f is an increasing function of X to Y if and only if it is an increasing function of X ′ to .
Y ′ Therefore, in the above theorem we may write lb in place of ub. However, because of Theorem 3.3 and Corollary 2.2, we cannot write sup instead of ub.
Despite this, in [40], we could also easily prove the following (1) f is increasing, To prove the implication (1) ⇒ (3), note that if (1) holds, then by the definition of maximum and Theorem 4.6 we have even if X is not assumed to be reflexive.

Closure and Interior Operations on Gosets
According to [40], we shall now also use the following Definition 5.1. If ϕ is a unary operation on a goset X, then we say that: ). ).
In this respect, it is also worth noticing that ϕ is upper (lower) semiidempotent if and only if its restriction to its range is extensive (intensive). Therefore, if ϕ is extensive (intensive), then ϕ is upper (lower) semiidempotent.
The importance of extensive functions is also quite obvious from the following theorem of [40].
The following theorem of [40] shows that, in contrast to the injective, increasing functions the inverse of an injective, extensive operation need not be extensive.
In general, the idempotent operations are quite different from the both upper and lower semiidempotent ones. However, we may still naturally use the following Definition 5.8. An increasing, extensive (intensive) operation is called a preclosure (preinterior) operation. And, a lower semiidempotent (upper semiidempotent) preclosure (preinterior) operation is called a closure (interior) operation.

Remark 5.9. Thus, ϕ is, for instance, an interior operation on a goset X if and only
if it is a closure operation on the dual X ′ of X.
Concerning closure operations, in [40] we have, for instance, proved the following two theorems.

sup sup
A A X Y ϕ = Remark 5.14. Surprisingly, the corresponding infimum properties of closure operations are much simpler, and do not require the transitivity of X.

The Induced Order and Interior Relations
Definition 6.1. For a function f of a set X to a goset Y, we define a relation f Ord on The relation f Ord will be called the natural order induced by f . Remark 6.2. Thus, under the notation , Therefore, as an immediate consequence of the corresponding definitions, we can state the following

Theorem 6.3. If f is a function of a set X to a goset Y, then f
Ord is the largest relation on X making f to be increasing.

Proof. If ≤ is a relation on X making f to be increasing, then
is also true.
Several further basic properties of the relation f Ord have been proved in [35] .
For instance, as some immediate consequences of some slightly more general results, we have established the following two theorems.

Theorem 6.4. If f is a function of a set X to a goset Y, then
Ord is a partial order on X if f is injective and Y is a poset.

Theorem 6.5. A function f of a goset X to a proset Y is increasing if and only if any
one of the following assertions holds: f Ord is ascending valued, 1 Ord − f is increasing, 1 Ord − f is descending valued.

Remark 6.6. A relation F on a goset X to a set Y is called increasing if the induced set-valued function
While, a relation F on a set X to a goset Y is called ascending valued if the function F is ascending valued. That is, for all X u ∈ and . Y w ∈ By [35], a relation F on a goset X to a set Y is increasing if and only if its inverse is ascending valued. And dually, a relation F on a set X to a goset Y is descending valued if and only if its inverse is decreasing.
1 Int − f is ascending valued.
Remark 6.13. If in particular X is also a goset and f is increasing, then we can also state that Int is descending valued.
However, in view of the corresponding results of Section 4, it is now more important to note that following theorem is also true.

Galois Type Connections between Gosets
In [35], slightly extending the ideas of Ore [16], Schmidt [20, p. 209], Blyth and Janowitz [2, p. 11], and the present author [33] on Galois connections, residuated mappings, and increasingly normal functions, we have introduced the following Definition 7.1. If f is a function on one goset X to another Y and g is a function of Y to X, then we say that: and . Y y ∈ Remark 7.2. Now, the function f may be naturally called increasingly g-normal if it is both increasingly upper and lower g-seminormal.
Moreover, a function f of X to Y may, for instance, be naturally called increasingly normal if it is increasingly g-normal for some function g of Y to X.
Later, we shall see that the increasingly normal functions are closely related to the increasing ones. Therefore, in accordance with Remark 4.2, a function f of X to Y may, for instance, be naturally called decreasingly normal if it is increasing normal as a function of X to the dual Y ′ of Y.
In this respect, it is also worth mentioning that in [35] we have proved the following simple dualization principle.

Theorem 7.3. If f is an increasingly upper (lower) g-seminormal function of one goset X to another Y, then g is an increasingly lower
Proof. If f is increasingly upper g-seminormal, then by the corresponding definitions it is clear that Therefore, g is increasingly lower f-seminormal as a function of Y ′ to . X ′

Corollary 7.4. If f is an increasingly g-normal function of one goset X to another Y,
then g is an increasingly f-normal function of Y ′ to . X ′ Remark 7.5. By Theorem 7.3, the properties of the functions g and g f can be immediately derived from those of f and . f g However, it is sometimes more convenient to apply a direct proof.
In [35], having in mind the properties of the function , f g = ϕ and slightly extending the ideas of Pataki [17] and the present author [33], we have also introduced the following Definition 7.6. If f is a function on one goset X to another Y and ϕ is a unary operation on X, then we say that: ( X v u ∈ Remark 7.7. Now, the function f may be naturally called increasingly ϕ -regular if it is both increasingly upper and lower ϕ -semiregular.
Moreover, a function f of X to Y may, for instance, be naturally called increasingly regular if it is increasingly ϕ -regular for some unary operation ϕ on X.
Analogously to Remark 7.2, a function f of X to Y may, for instance, be naturally called decreasingly regular if it is increasingly regular as a function of X to . Y ′ Unfortunately, now we do not have a counterpart of Theorem 7.3.
However, to clarify the relationship between normal and regular functions, in [35] we have proved the following two theorems.
Theorem 7.8. If f is an increasingly upper (lower) g-seminormal function of one goset X to another Y, then f g = ϕ is a unary operation on X such that f is increasingly upper (lower) ϕ -semiregular.

Corollary 7.9. If f is an increasingly g-normal function of one goset X to another Y,
is a unary operation on X such that f is increasingly ϕ -regular.
Theorem 7.10. If f is an increasingly upper (lower) ϕ -semiregular function of one goset X onto another Y and g is a function of Y to X such that , Proof. Suppose that X x ∈ and . Y y ∈ Then, since [ ], Now, if f is increasingly upper ϕ -semiregular, then we can easily see that Therefore, f is increasingly upper g-seminormal too.

Corollary 7.11. If f is an increasingly ϕ -regular function of one goset X onto another Y and g is a function of Y to X such that
, Remark 7.12. By Theorem 7.8, it is clear that several properties of the increasingly normal functions can be immediately derived from those of the increasingly regular ones. Therefore, the latter ones have to study before the former ones.
Moreover, from Theorem 7.10, we can see that the increasing regular functions are still less general objects than the increasingly normal ones. Later, we shall see that they are strictly between closure operations and increasingly normal functions.

Some Basic Properties of Increasingly Semiregular Functions
In [35], as some immediate consequences of the corresponding definitions, we have also proved the following theorems and their corollaries. Proof. Because of the reflexivity of Y, for any , X x ∈ we have ( ) ( ).
x f x f ≤ Hence, by using the assumed semiregularity of f, we can already infer that ( ).
x x ϕ ≤ Therefore, ϕ is extensive, and thus by Remark 5.3 it is also upper semiidempotent.

Corollary 8.2. If f is an increasingly upper ϕ -semiregular function of an arbitrary goset X to a reflexive one Y such that f is increasing, then
. ϕ ≤ f f Theorem 8.3. If f is an increasingly lower ϕ -semiregular function of a reflexive goset X to an arbitrary one Y, then . f f ≤ ϕ Corollary 8.4. If f is an increasingly ϕ -regular function of a reflexive goset X to a reflexive, antisymmetric one Y such that f is increasing, then . ϕ = f f Theorem 8.5. If f is an increasingly ϕ -regular function of a reflexive goset X to a transitive one Y, then ϕ is lower semiidempotent.

Characterizations of Increasingly Seminormal Functions
The following theorems and their corollaries have also been proved in [35].

Theorem 11.1. For a function f of one goset X to another Y and a function g of Y to
X, the following assertions are equivalent: (2) f is an increasingly lower g-seminormal.
Theorem 11.2. For a function f of one goset X to another Y and a function g of Y to X, the following assertions are equivalent: (3) f is an increasingly upper g-seminormal.

Corollary 11.3. For a function f of one goset X to another Y and a function g of Y to
X, the following assertions are equivalent: (1) f is an increasingly g-normal, (2) then f is increasingly lower g-seminormal.
Corollary 11.8. For an increasing function f on a reflexive goset X to a transitive one Y and an arbitrary function g of Y to X, the following assertions are equivalent: for all , Y y ∈ (2) f is increasingly lower g-seminormal.
Theorem 11.9. For a function f of one proset to another Y and a function g of Y to X, the following assertions are equivalent: (1) f is increasingly g-normal, (2) f is increasing and ( ) ( ( )) y y g f Int max ∈ for all . Y y ∈ Remark 11.10. Note that if in particular X is a poset, then by Theorem 3.5 we may for all Y y ∈ in assertion (2).
From the above results, we can immediately derive several characterizations of upper and lower semiinvolutive operations.
For instance, from Theorem 11.9, by using Theorems 9.16 and 10.9, we can immediately derive the following Theorem 11.11. For an increasing operation ϕ on proset X, the following assertions are equivalent: (2) ϕ is both upper and lower semiinvolutive.  (2) we may simply write that ϕ is involutive.
From the above results, by using the Axiom of Choice, we can also immediately derive several useful characterizations of increasingly upper and lower seminormal functions.
For instance, from Theorem 11.9 we can immediately derive the following Theorem 11.13. For a function f of one proset X to another Y, the following assertions are equivalent: (1) f is increasingly normal, (2) f is increasing and Hence, it is clear that in particular we also have Corollary 11.14. For a function f of a max-complete proset X to an arbitrary one Y, the following assertions are equivalent: (1) f is increasingly normal, Now, more specially we can also state Corollary 11.15. For a function f of a max-complete proset X onto an arbitrary one Y, the following assertions are equivalent: (1) f is increasing, (2) f is increasingly normal.

Some Further Characterizations of Increasingly Normal Functions
In this section, we shall extend the results of [31, Section 7] to the present more general setting of increasingly normal functions.
For this, it is convenient to start with the following striking property of increasingly normal functions which also fails to hold for the increasing ones. ).

Directions to Some Further Reasonable Investigations
Some results of this paper can also be generalized to relator spaces of the form where X and Y are sets and R is a relator (family of relations) on X to Y. (For the origins of these concepts, see [22] and the references therein.) Note that relator spaces of the simpler type are already substantial generalizations of ordered sets [7] and uniform spaces [9]. However, they are insufficient to include context spaces [10, p. 17] (which are simple relator spaces of the where R is a relation on X to Y), and also to naturally express continuity properties of functions and relations on one relator space to another [26,37,50]. ).
Lb Lb R S = For this, it is enough to use only some basic facts on increasingly regular functions of one power set to another [17,30] The latter equality, proved first in [27], establishes a similar connection between analysis and algebra as the famous Euler formula does between exponential and trigonometric functions [21, p. 227] .
From the results of [30], it has become completely clear that, to unitedly treat the several basic structures derived from relators [27] and their associated closure and modification operations [37], it is necessary to investigate first increasingly seminormal and semiregular functions of one power set to another. Therefore, these functions have to be studied at least three stages of generality. Firstly for posets, secondly for power sets, and thirdly for relator spaces.
To study increasingly normal and regular function of one power set ( ) X P to a goset Y, in an immediate continuation of this paper, for a function F on a power set ( ) X P to a goset Y we shall carefully investigate the set-valued functions F G and , for all Y y ∈ and . X A ⊆ (Note that, here by identifying singletons with their elements, we have again written x in place of { } x for all . X x ∈ ) Thus, we also have Moreover, we may naturally look for some necessary and sufficient conditions in order that F could be increasingly upper and lower F G -seminormal and F Φ -semiregular, respectively.