Table Of Content(cid:3)
Lagois Connections
{ a Counterpart to Galois Connections {
Austin Melton Bernd S. W. Schr(cid:127)oder
Department of Computer Science Department of Mathematics
Michigan Technological University Hampton University
Houghton, Michigan 49931, USA Hampton, Virginia 23668, USA
George E. Strecker
Department of Mathematics
Kansas State University
Manhattan, Kansas 66506, USA
December 23, 1993
Abstract
In this paper we de(cid:12)ne a Lagois connection, which is a generaliza-
tion of a special type of Galois connection. We begin by introducing
two examples of Lagois connections. We then recall the de(cid:12)nition of
Galois connection and some of its properties; next we de(cid:12)ne Lagois
connection, establish some of its properties, and compare these with
propertiesof Galois connections; and thenwe (further)develop exam-
ples of Lagois connections. Via these examples it is shown that, as is
the case of Galois connections, there is a plethora of Lagois connec-
tions. Alsoitisshownthatseveralfundamentalsituationsincomputer
scienceandmathematicsthatcannotbeinterpretedintermsofGalois
connections naturally (cid:12)t into the theory of Lagois connections.
key words: Galois connection, Galois insertion, Lagois connection, quasi-
inverse, poset system, closure operator, interior operator
AMS subject classi(cid:12)cation:
Primary: 06A15, 06A10
Secondary: 68F05, 68F99, 54B99
(cid:3)
TheauthorswerepartiallyfundedbythetheO(cid:14)ceofNavalResearchunderContract
N00014-88-K-0455.
1
1 Introduction
A Galoisconnection is an elegant and easily de(cid:12)ned relationshipamong pairs
of partially ordered sets and order preserving maps between them. In [6] it
is shown that some activities which commonly occur in computer science are
examples of Galois connections; these examples include showing correctness
ofatranslatorandde(cid:12)ningacoercionmapbetween datatypes. However, the
value of Galois connections can not be fully appreciated simply by studying
a few examples. Galois connections are important because they occur very
commonly and because when a particular situation is a Galois connection
then Galois connection results can often be used to make the situation much
more easily understood. Though sometimes the existence of a Galois con-
nection has a major impact on proving results about a given situation, it is
moreoften the case that Galoisconnection results make characteristics of the
situation very organized and clear. Thus, for a theoretical computer scientist
or a mathematician it is perhaps helpful to know of some examples of Galois
connections, but it is probably even more important to know what Galois
connections are and to know some of their properties. Similar remarks are
true for the new Lagois connection de(cid:12)ned in this paper.
It is of course also true that the properties of Galois connections and
Lagois connections are themselves intrinsically interesting, especially when
the simplicity of the de(cid:12)nitions are compared with the (number of) results
that follow from them.
If one looks atthe computer science examples of Galoisconnections in[6],
itisinteresting tonote that allthe examples are infactGaloisinsertions, i.e.,
the residuated or lower adjoint part of the connection is always one-to-one.
One is of course challenged to see what these examples become when they
are generalized, and here's where this paper really starts. These generalized
examples are not Galois connections; they are Lagois connections.
A Galois connection can also be viewed as a closure operator on one of its
partiallyordered sets and an interior operator on the other one, with the sets
of closed points and open points being isomorphic partially ordered sets (cf.
Corollary 1.5). However, when we generalize the translator and coercion ex-
amples of [6], we obtaininone case two closure operators whose sets of closed
points are isomorphic and in the other case two interior operators whose sets
of open points are isomorphic, i.e., the generalizationsare Lagois connections
(cf. Corollary 3.21). See Figure 1 for generic diagrams of a Galois connec-
2
tion and a Lagois connection involving two closure operators. The diagrams,
we think, show why the name Lagois seems appropriate. However, from the
diagrams or the name one should not conclude that Lagois connections are
simply a minor modi(cid:12)cation of Galois connections. Though some Lagois
connections are special Galois connections (and some Galois connections are
special Lagois connections, see Subsection 3.4), (general) Galois connections
and (general) Lagois connections model fundamentally di(cid:11)erent kinds of sit-
uations, i.e., the existence of two suitably linked closure operators in both
partiallyordered sets or two similarlylinked interior operators inboth is fun-
damentally di(cid:11)erent than the one closure operator and one interior operator
that are linked in a Galois connection.
In [6] it is shown that when verifying that a translation from a source
language to a target language is correct it is su(cid:14)cient to show that the
translation is the residuated (or lower adjoint) part of a Galois insertion
which is a special Galois connection. However, in trying to apply these
results to a generalized situation, we run into problems. It turns out that
a Galois insertion is a special Galois connection and also a special Lagois
connection, and the appropriate generalization of the above example is a
Lagois connection that is not a Galois connection.
The dual symmetry between closure and interior operators inherent in
Galois connections helps give them their elegant simplicity. Lagois connec-
tions have an entirely di(cid:11)erent kind of symmetry, one that is also relevant in
computer science and mathematical examples.
Although many of the properties of Lagois connections are direct ana-
logues of corresponding properties of Galois connections (compare, e.g.,
1.2 { 1.4 with 3.5 { 3.13), the lack of the dual symmetry means that some
properties of Lagois connections are destined to be quite di(cid:11)erent from those
of Galois connections (compare 1.2(9) and 1.6 with 4.10 and 4.9). However,
Lagois connections are found in abundance, and the recognition that an en-
tity is a Lagois connection can make understanding it and working with it
much easier.
Another example discussed in [6] is a coercion map between the Boolean
values with a bottom element and the natural numbers with a bottom ele-
ment. In both sets the ordering for non-bottom elements is equality, i.e., if
two elements are not equal and neither is the bottom element then they are
not comparable. In [6] the coercion map is again part of a Galois insertion,
(and in fact part of a special Lagois connection), and when we generalize this
3
example we again arrive at a general Lagois connection that is not a Galois
connection. In fact, in the generalization obtained the Lagois connection is
the only possible Galois or Lagois connection between the given partially
ordered sets for which \False", \Neutral", and \True" are carried to (cid:0)1, 0,
and 1, respectively (see Example 4.2).
Below (for comparison purposes) we (cid:12)rst recall the de(cid:12)nition of Galois
connection and some of its properties. In section 2 we introduce Lagois con-
nections and in section 3 we establish some of their properties and compare
these with the properties of Galois connections. Section 4 contains examples
of Lagois connections in both computer science and mathematics.
1.1 Galois connections
De(cid:12)nition 1.1
1. If (P;(cid:20)) and (Q;(cid:20)) are partially ordered sets, and f : P ! Q and
g: Q ! P are order preserving functions, then we call the quadruple
((P;(cid:20));f;g;(Q;(cid:20))) or simply (P;f;g;Q) a poset system.
2. A poset system ((P;(cid:20));f;g;(Q;(cid:20))) is called a Galois connection
provided that
(GC1) gf is an increasing function, i.e., gf(p) (cid:21) p for all p 2 P, and
(GC2) fg is a decreasing function, i.e., fg(q) (cid:20) q for all q 2 Q.
The function f is called a residuated (or lower adjoint) map and
the function g is called a residual (or upper adjoint) map. The Galois
connection is called a Galois insertion i(cid:11) f is one-to-one or, equivalently,
i(cid:11) g is onto (cf. Proposition 1.2(13).)
1.2 Facts about Galois connections
The following results are well-known. Proofs of them are given in [2], [3], [6],
[7], and [8]. In that which follows we will show that results similar to many
of them hold for Lagois connections but that for others no reasonable Lagois
analogue is available.
Proposition 1.2 Let ((P;(cid:20));f;g;(Q;(cid:20))) be a Galois connection. Then
4
1. g is a quasi-inverse for f, i.e., fgf = f, and f is a quasi-inverse for
g, i.e., gfg = g.
2. ((Q;(cid:21));g;f;(P;(cid:21))) is a Galois connection.
3. The images g[Q] and f[P] are isomorphic partially ordered sets, and
the restrictions of f and g to these images are isomorphisms that are
inverses of each other.
4. For each p 2 P, p 2 g[Q] i(cid:11) gf(p) = p; and for each q 2 Q, q 2 f[P]
i(cid:11) q = fg(q).
5. gf isaclosureoperatoronP (i.e., it isidempotent, order-preserving,
and increasing); and fg is an interior operator on Q (i.e., it is
idempotent, order-preserving, and decreasing).
(cid:0)1
6. For each q 2 f[P]; f (q) has a largest member, which is g(q); and
(cid:0)1
for each p 2 g[Q]; g (p) has a smallest member, which is f(p).
V
(cid:3) (cid:3)
7. For each p 2 P; gf(p) = fp 2 g[Q]jp (cid:21) pg and for each q 2 Q;
W
(cid:3) (cid:3)
fg(q) = fq 2 f[P]jq (cid:20) qg.
8. The functions f and g uniquely determine each other; in fact f(p) =
V W
fq 2 Q j p (cid:20) g(q)g and g(q) = fp 2 P j f(p) (cid:20) qg.
9. f preserves joins, and g preserves meets.
10. If A (cid:18) g[Q], then the meet of A in g[Q] exists if and only if the meet
of A in P exists, and whenever either exists, they are equal; and if
B (cid:18) f[P], then the join of B in f[P] exists if and only if the join of B
in Q exists, and whenever either exists, they are equal.
11. If P has ((cid:12)nite) joins, then so does g[Q], but these might not coincide
with the joins in P, and if Q has ((cid:12)nite) meets, then so does f[P], but
these might not coincide with the meets in Q; in particular the join a^
0
of A (cid:18) g[Q] in g[Q] exists if the join a of A in P exists, and in this
0 ^
case a^ = gf(a), and the meet b of B (cid:18) f[P] in f[P] exists if the meet
0 ^ 0
b of B in Q exists, and in this case b = fg(b).
12. If P and Q are (complete) lattices, then so are g[Q] and f[P]; but they
need not be sublattices of P or Q.
5
13. f is one-to-one i(cid:11) g is onto i(cid:11) gf = idP; and g is one-to-one i(cid:11) f is
onto i(cid:11) fg = idQ.
Theorem 1.3 Let P and Q be posets. Then an order-preserving function
f : P ! Q is a lower adjoint of a Galois connection i.e., it has an upper
adjoint g (such that (P;f;g;Q) is a Galois connection) i(cid:11) for all q 2 Q, there
(cid:0)1
is some p 2 P such that f (#q) = #p, i.e., i(cid:11) the inverse image under f of
every principal ideal in Q is a principal ideal in P.
Theorem 1.4 Let (P;(cid:20)) and (Q;(cid:20)) be posets. There is a Galois connection
between (P;(cid:20)) and (Q;(cid:20)) if and only if the following four conditions hold:
(cid:3) (cid:3) (cid:3) (cid:3)
1. There exist P (cid:18) P, Q (cid:18) Q, and an order-isomorphism i: P ! Q .
(cid:3)
2. There exists an equivalence relation (cid:24)P on P such that P is a system
of representatives for (cid:24)P and there exists an equivalence relation (cid:24)Q
(cid:3)
on Q such that Q is a system of representatives for (cid:24)Q. The members
(cid:3) (cid:3)
of P resp. Q are called the buds or budpoints and the equivalence
classes are called blossoms.
(cid:3) (cid:3) (cid:3) (cid:3)
3. If p 2 P and p 2 P with p (cid:24)P p , then p (cid:20) p ; and if q 2 Q and
(cid:3) (cid:3) (cid:3) (cid:3)
q 2 Q with q (cid:24)Q q , then q (cid:21) q .
(cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
4. If p1 (cid:20) p2 in P and p1;p2 2 P with p1 (cid:24)P p1 and p2 (cid:24)P p2, then
(cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)
p1 (cid:20) p2, and if q1 (cid:20) q2 in Q and q1;q2 2 Q with q1 (cid:24)Q q1 and
(cid:3) (cid:3) (cid:3)
q2 (cid:24)Q q2, then q1 (cid:20) q2.
(cid:3) (cid:3)
If the four conditions hold and r: P ! P and s: Q ! Q are the canonical
mappings onto the systems of representatives, then the Galois connection is
(cid:0)1
given by (P;ir;i s;Q).
Corollary 1.5 Let P and Q be posets, c : P ! P be a closure operator, and
i : Q ! Q be an interior operator such that c[P] and i[Q] are isomorphic
(with their inherited orders). If h : c[P] ! i[Q] is such an isomorphism, then
(cid:0)1
(P;hc;h i;Q) is a Galois connection.
Theorem 1.4 and Theorem 3.20 (see below) give an insight in the struc-
ture of Galois and Lagois connections, and make it possible to draw an easy-
to-conceive picture that indicates these structures and their di(cid:11)erences (see
Figure 1 ).
6
(cid:7)(cid:4)
(cid:3)(cid:0) (cid:3)(cid:0)
DD (cid:4)(cid:4)
D(cid:4) D(cid:4)
r r r r r D(cid:4)r r D(cid:4)r D(cid:4)r r
(cid:7)(cid:4)
(cid:4)DL (cid:12)(cid:4)D (cid:4)D (cid:12) L (cid:12) (cid:12)
(cid:6)(cid:4)(cid:4) DDL(cid:5)L(cid:12)r(cid:12)(cid:2)(cid:4)D(cid:1)(cid:6)(cid:4)(cid:4)(cid:12)rDD(cid:12)(cid:5)(cid:12) LL(cid:12)r(cid:12)(cid:7)(cid:4)DDD(cid:12)(cid:4)r(cid:4)(cid:12)(cid:4)(cid:12)
(cid:4)LD (cid:4)D L
(cid:4)(cid:4) DDL (cid:12)(cid:2)(cid:4)D(cid:1) (cid:27) g LDD (cid:4)(cid:4)(cid:12)
P (cid:6)(cid:5)Lr(cid:12) - LD(cid:4)r(cid:12) Q
(cid:12) (cid:12)
(cid:12) f (cid:12)
r(cid:12) r(cid:12)
(cid:12)(cid:4)DL (cid:3)(cid:0)(cid:12)L (cid:3)(cid:0)
r(cid:12)(cid:12)(cid:6)(cid:4)(cid:4) DDL(cid:5)Lr DD(cid:4)(cid:12)r(cid:4)(cid:12) LLDD(cid:4)r(cid:4)
(cid:4)D
(cid:4)D
(cid:2)(cid:1)
a) Galois connection: The blossoms in Q are growing upwards
r r r r r r r r r r
(cid:4)DL (cid:12)(cid:4)D (cid:12)(cid:4)D (cid:4)DL (cid:12) (cid:4)D (cid:12)(cid:4)D
(cid:2)(cid:4)D(cid:1)LL(cid:12)r(cid:12)(cid:6)(cid:4)(cid:4) DD(cid:5)r(cid:12)(cid:12)(cid:6)(cid:4)(cid:4) DD(cid:5) (cid:6)(cid:4)(cid:4) DDL(cid:5)L(cid:12)r(cid:12) (cid:2)(cid:4)rD(cid:12)(cid:1)(cid:12)(cid:2)(cid:4)D(cid:1)
(cid:12) (cid:12)
(cid:4)LD (cid:4)D (cid:4)LD (cid:4)D
(cid:2)(cid:4)D(cid:1)L (cid:12)(cid:2)(cid:4)D(cid:1) (cid:27) g (cid:2)(cid:4)D(cid:1)L (cid:12)(cid:4)(cid:4) DD
P Lr(cid:12) - Lr(cid:12)(cid:6)(cid:5) Q
(cid:12) (cid:12)
(cid:12) f (cid:12)
r(cid:12) r(cid:12)
(cid:12)L (cid:12)(cid:4)DL
(cid:12) L (cid:12)(cid:4)(cid:4) DDL
r(cid:12) Lr (cid:12)r (cid:6)(cid:5)Lr
(cid:4)D (cid:4)D (cid:4)D
(cid:2)(cid:4)D(cid:1) (cid:2)(cid:4)D(cid:1) (cid:2)(cid:4)D(cid:1)
b) Increasing Lagois connection: The blossoms in Q are growing downwards
Figure 1: The structure of Galois and Lagois connections
7
^
Proposition 1.6 If (P;f;g;Q) and (Q;f;g^;R) are Galois connections, then
^
so is (P;ff;gg^;R).
^
We will denote the Galois connection (P;ff;gg^;R) of the above propo-
^
sition by (Q;f;g^;R) (cid:14)(P;f;g;Q), and call it the composite of (P;f;g;Q)
^
with (Q;f;g^;R). This composition operation is clearly associative.
Proposition 1.7 Every Galois connection (P;f;g;Q) is a composite of Ga-
lois connections
0 0 0 0
(Q;e2;r2;Q)(cid:14)(P ;i1;i2;Q)(cid:14)(P;r1;e1;P )
where
(1) r1 is onto and e1 is one-to-one,
(2) i1 and i2 are isomorphisms that are inverse to each other, and
(3) e2 is one-to-one and r2 is onto.
2 Lagois Connections
In considering poset systems from computer science and mathematics, one
frequently has the situation of a system ((P;(cid:20));f;g;(Q;(cid:20))) for which both
fg and gf are closure operators (resp., both are interior operators). The
existence of such systems motivates the concept of \Lagois connection".
These new connections, then, come in two types which we call increasing
(or closed) Lagois connections and decreasing (or open) Lagois connections.
De(cid:12)nition 2.1 A poset system L = (P;f;g;Q) is called an increasing
(or closed) Lagois connection i(cid:11)
(LC1) gf is an increasing function,
(LC2) fg is an increasing function,
(LC3) fgf = f, and
(LC4) gfg = g.
L is called a decreasing (or open) Lagois connection i(cid:11) both gf and
fg are decreasing and fgf = f and gfg = g.
8
Notice that the (cid:12)rst two conditions((LC1) and (LC2)) inthe de(cid:12)nitionof
Lagois connection are just like (GC1) and (GC2) in the de(cid:12)nition of Galois
connection, but with (LC2) \switched". The last two conditions ((LC3) and
(LC4)) that each of f and g is a quasi-inverse for the other are precisely part
(1) of Proposition 1.2. Thus every Galois connection satis(cid:12)es three of the
four de(cid:12)ning properties of a Lagois connection. Unlike the case of Galois
connections, these last two conditions do not follow from the (cid:12)rst two [see
Example 4.8 and Proposition 3.15], and without them the resulting concept
appears to be too weak to be of general interest.
3 Results
3.1 Lagois duality and symmetry
De(cid:12)nition 3.1 If J = ((P;(cid:20));f;g;(Q;(cid:20))) is a poset system then the poset
system ((P;(cid:21));f;g;(Q;(cid:21))) will be called the opposite or dual of J and will
op
be denoted by J , whereas ((Q;(cid:20));g;f;(P;(cid:20))) will be called the transpose
tr
of J and will be denoted by J .
Proposition 3.2 For any poset system J = ((P;(cid:20));f;g;(Q;(cid:20))) the follow-
ing are equivalent:
1. J is an increasing Lagois connection.
tr
2. J is an increasing Lagois connection.
op
3. J is a decreasing Lagois connection.
op tr
4. (J ) is a decreasing Lagois connection.
By the above proposition, increasing and decreasing Lagois connections
can be easily transformed into each other, and statements about one type
of Lagois connection can be immediately transformed into statements about
the other. Therefore in the remainder of the paper we will deal primarily
with increasing Lagois connections, and we will call them simply Lagois
connections when no confusion is likely. The proposition above gives an
analogueof Proposition 1.2(2), which (with the above terminology)says that
op tr
a poset system J is a Galois connection if and only if (J ) is a Galois
op tr
connection. Note that if J is a Galois connection then neither J nor J
need be Galois connections.
9
3.2 Properties of Lagois Connections
Since the following proposition holds for all poset systems, it provides some
insight into Proposition 1.2(3) and yields the analogous result for Lagois
connections. Its corollary also provides an alternative de(cid:12)nition for Lagois
connection.
Proposition 3.3 Let J = (P;f;g;Q) be a poset system.
(1) J satis(cid:12)es the condition (LC3) (i.e., fgf = f) if and only if it satis(cid:12)es
the condition
(LC3') for each q 2 Q, q 2 f[P] if and only if fg(q) = q.
(2) J satis(cid:12)es the condition (LC4) (i.e., gfg = g) if and only if it satis(cid:12)es
the condition
(LC4') for each p 2 P, p 2 g[Q] if and only if gf(p) = p.
Corollary 3.4 A poset system (P;f;g;Q;) is a Lagois connection if and
only if it satis(cid:12)es properties (LC1), (LC2), (LC3'), and (LC4').
The next six results are direct analogues of 1.2(3), 1.2(5) through 1.2(8),
and 1.3.
Proposition 3.5 Let (P;f;g;Q) be a Lagois connection. Then the images
g[Q] and f[P] are isomorphic partially ordered sets, and the restrictions of
f and g to these images are isomorphisms that are inverses of each other.
Proposition 3.6 Let (P;f;g;Q) be a Lagois connection. Then both gf and
1
fg are closure operators.
Proposition 3.7 Let (P;f;g;Q) be a Lagois connection and let q 2 f[P]
(cid:0)1 (cid:0)1
and p 2 g[Q]. Then f (q) has a largest member, which is g(q), and g (p)
has a largest member, which is f(p).
1
Note that for decreasing (or open) Lagois connections both gf and fg are interior
operators.
10
