The Structure of Interlaced Bilattices
The Structure of Interlaced Bilattices Arnon Avron Department of Computer Science Sackler Faculty of Exact Sciences Tel Aviv University, Ramat Aviv, Israel 69978 Abstract
Bilattices were introduced and applied by Ginsberg and Fitting for a diversity of applications, such as truth maintenance systems, default inferences and logic programming. In this paper we investigate the structure and properties of a particularly important class of bilattices called interlaced bilattices (introduced by Fitting). The main results are that every interlaced bilattice is isomorphic to the Ginsberg-Fitting product of two bounded lattices and that the variety of interlaced bilattices is equivalent to the variety of bounded lattices with two distinguishable distributive elements which are complements of each other . This implies that interlaced bilattices can be characterized using a nite set of equations. Our results generalize to interlaced bilattices results of Ginsberg, Fitting and Jonsson for distributive bilattices.
Introduction The notion of a bilattice was introduced by Ginsberg in [Gi88] as a general framework for a diversity of applications, such as truth maintenance systems, default inferences and others. The notion was further investigated and applied for logic programming by Fitting ([Fi89], [Fi90], [Fi91], [Fi94]). The main idea behind bilattices is to use structures in which there are two partial order relations, having di erent interpretations. There should, of course, be a connection between the two relations. Ginsberg uses for this an operation of negation which is order preserving w.r.t. one, an involution w.r.t. to the other. Another connection he considered is distributive laws (12 altogether). This was generalized by Fitting who introduced the notion of an interlaced bilattice, in which all the basic bilattice operations are order preserving w.r.t. both partial orders. The structure of distributive bilattices is well understood since Ginsberg and Fitting proved a characterization theorem for them, to the e ect that each such bilattice is isomorphic to a certain product of two distributive bounded lattices. Fitting has further observed that if we apply that construction to any two bounded lattices (not necessarily distributive) 1
we get an interlaced bilattice. The converse, however, was not established, and the structure of interlaced bilattices has not, so far, been well understood. In this paper we thoroughly investigate the structure of interlaced bilatttices (with and without negation), and their important properties. Our main result is a generalization to interlaced bilattices of the Ginsberg-Fitting characterization of distributive bilattices (i.e. the converse of Fitting's observation). Other central results are characterizations using equational bases and the equivalence of the variety of interlaced bilattices with the variety of bounded lattices with two complementary, distributive elements. (This generalizes to interlaced bilattice results of Jonsson in [Jo94] for distributive bilattices.)
Note In order to make the presentation complete and self-contained, we repeat, together with the appropriate references, some short proofs which have already appeared elsewhere.
1 General Background
1.1 De nition [Fi90]
An interlaced bilattice (IBL) is a structure = B; ; ; ; ; ; ; t; f; ; B
h
t k ^ _
> ?i
such that
1. B; ; ; ; t; f is a bounded lattice (with the order relation, and the meet and join operations, and t, f the maximal and minimal elements, respectively). h
t ^ _
i
2. B; ; ; ; ; h
k > ?i
t
^
_
is also a bounded lattice.
3. Each of the four operations ; ; ; is order preserving with respect to both (e.g. if a b then a c b c). _ ^
k
k
^
k
t
and
^
Note In the original de nitions of Ginsberg and Fitting B; and b; are required to be complete lattices. We shall call below IBLs with this property complete. h
t i
h
k i
1.2 De nition [Gi88] An IBL is called distributive if all the twelve possible distributive laws concerning ; ; and hold.
^ _
2
Note Fitting has observed (see, e.g. [Fi90]) that the distributive laws imply condition (3) of 1.1. For example if a b then b = a b, and so c b = c (a b) = (c a) (c b) by the distributive laws. Hence c a c b in this case. t
_
t
_
_
1.3 De nition A unary operation on an IBL is called negation if it is order-preserving with respect to and an involution w.r.t. . In other words, is negation if
B
k
t
a=a a b b a b a
(i) (ii) (iii)
t
)
k
)
a b:
t
t
Notes 1. Ginsberg's original de nition of a bilattice in [Gi88] is: a structure which satis es the rst two conditions in de nition 1 (+ completeness) and has a negation. 2. Obviously, if is a negation then (a b) = a a b, (a b) = a b, t = f , f = t,
^
b, (a b) = a
_
>
_
= , >
?
= .
b, (a b) =
^
?
The main method of constructing an IBL is described in the following de nition. It was essentially introduced by Ginsberg in [Gi88], and further generalized by Fitting in [Fi90].
1.4 De nition Let = L; ; ; ; 1 ; 0 and = R; ; ; ; 1 ; 0 be two bounded lattices. Their product, , is the structure L R; ; ; ; ; ; ; t; f; ; de ned as follows: L
h
L tL uL
Li
L
R
L R
(i) (ii) (iii) (iv) (v) (vi) (vii)
h
h
R tR uR
Ri
R
t k ^ _
> ?i
(a ; b ) (a ; b ) a a and b b (a ; b ) (a ; b ) a a and b b (a ; b ) (a ; b ) = (a a ; b b ) (a ; b ) (a ; b ) = (a a ; b b ) (a ; b ) (a ; b ) = (a a ; b b ) (a ; b ) (a ; b ) = (a a ; b b ) t = 1 ;0 f = 0 ;1 = 1 ;1 1
1
k
2
2
,
1 L
2
1 R
2
1
1
t
2
2
,
1 L
2
1 R
2
1
1
_
2
2
1 tL
2
1 uR
2
1
1
^
2
2
1 uL
2
1 tR
2
1
1
2
2
1 tL
2
1 tR
2
1
1
2
2
1 uL
2
1 uR
2
Ri
>
h
L
Ri
h
L
3
h
L
Ri
?
= 0 ;0 h
L
Ri
1.5 Theorem 1. [Fi90,91], [Gi88] L R is an IBL. Moreover, if both L and R are distributive lattices then so is L R, and if both L and R are complete lattices then so is L R. 2. [Gi88] If L is a bounded lattice then the operation , de ned by (x; y) = (y; x), is a negation on L L.
The proof of this theorem is straightforward. For distributive bilattices, a converse to Theorem 1.5 was proved by Ginsberg and Fitting (in [Gi88], [Fi90]): if is a distributive bilattice then there exist distributive bounded lattices and such that is isomorphic to . These and are unique (up to isomorphism). Ginsberg-Fitting's result will easily follow from a generalization we prove below for every IBL. B
L
R
B
L R
L
R
2 Basic Properties of IBLs In this section is a xed interlaced bilattice. B
2.1 De nition [a; b] = x B a [a; b] = x B a
(1) (2)
t
f
2
j
t
f
2
j
k
k
x b x b t
k
g g
2.2 Proposition 1. If a
t
b then [a; b] = [a b; a b] .
2. If a
k
b then [a; b] = [a b; a b] .
t
^
k
k
_
t
Proof We show the rst formula as an example. Assume rst that y [a; b] , so a y b. Since is interlaced, a (a b) y (a b) b (a b). Hence a b y (a b) a b and so y (a b) = a b. It follows that a b y. Similarly y a b. Hence y [a b; a b] . For the converse, assume that a b and a b y a b. Then a (a b) a y a (a b). But, if a b then a = a a a b and so a (a b) = a. Similarly, a (a b) = a. It follows that a a y a and so a y = a and a y. Similarly, y b and so y [a; b] . 2 2
B
t
t
k
^
t
^
k
2
t
k
k
t
^
^
k
t
4
t
k
k
t
t
t
2
t
^
^
^
t
t
k
k
2.3 Corollary [a; b] and [a; b] are closed under ; ; and . Moreover, in case a b then [a; b] is an interlaced bilattice, with the same order relations and bilattice operations as (but with a; b; a b; a b taking the roles of f; t; ; respectively). Similarly, if a b then [a; b] is an IBL with the same order relations and bilattice operations as . t
^ _
k
t
t
B
? >
k
k
B
2.4 Corollary If a
b
t
b then b f a a b and a
^ ? t
t
k
_?
k
b f and a t b a .
^ > t
t
k
b
k
a t. Similarly, if a
t
b then a [f; b] and b [a; t] .
k
b then
_ >
Proof The rst part follows from 2.2 and the fact that if a Similar considerations apply for the second part.
2
2
t
t
2.5 Corollary [Fi90] t f= ; =f ;
(i) (ii)
t f=
?
?^>
?_>
>
=t
Proof Since B = [f; t] , we have by 2.2 that B = [f t; f t] . But also B = [ ; ] . Hence, the two equations in (i). The proof of (ii) is similar.
t
? >
k
k
2.6 Proposition x x
(i) (ii)
^? ^>
=x f ; =x f ;
x x
_? _>
=x t; =x t:
Proof We show the rst equality. The proof of the rest is similar. Since x x we have by 2.4 that x f x . To show the converse it is enough to show that x x and x f (since is the meet operation of ). The rst inequality follows from x, since x x x x = x. The second inequality follows from f , since f x x f = f. 2 ^ ? t
^ ? k
^ ? k
? k
? k
k
? k
? k
)
^?
)
^ ? k
5
^ ? k
^
k
^
Notes 1. For distributive bilattices 2.6 was shown in [Jo94]. 2. If we substitute, e.g. for x in the rst equality we get another proof of Hence 2.5 is a corollary of 2.6. >
>^ ?
= f.
2.7 Proposition If x If x If x If x
(1) (2) (3) (4)
k
k
t
t
b then x = (x b then x = (x b then x = (x b then x = (x
^ ^
b) b) b) b)
_ ^
(x (x (x (x
_ _
b) b) b) b)
Proof Again we prove only (1). Assume x b. Then x x x b and so x x b. Similarly, x x b. By combining these two inequalities we get x (x b) (x b). On the other hand x [x b; x b] , and so , by 2.2, x [(x b) (x b); (x b) (x b)] . Hence, x (x b) (x b), and the quality follows. 2 k
^
k
k
2
^
^
k
^
_
^
^
k
_
_
2
_
k
k
^
^
_
_
2.8 Corollary (1) (2) (3) (4)
x = (x ) (x ) = (x f ) (x t) ; x = (x ) (x ) = (x f ) (x t) ; x = (x f ) (x f ) = (x ) (x ) ; x = (x t) (x t) = (x ) (x ) : ^?
_?
^>
_>
_
^
^?
_ ?
_
^
^>
_ >
Proof Immediate from 2.7 and 2.6.
Note For distributive bilattices 2.8 is an immediate corollary of 2.5 and 2.6 (an so { of 2.6) We end this section with a result which was shown for distributive bilattices in [Jo94]. 6
t
2.9 Proposition (1) (2)
k k
= =
t
t
i f = ? i t = > i f = > i t = ?
Proof 1. Obviously, if = then f = and t = . For the converse, assume, e.g. that f = . Then, for all a; b: k
b
t
a
,
t
b [f; a] 2
?
b [ ; a]
,
t
>
2 ?
t
2:2 ,
?
b [
2 ?
a;
?
a]
k
,
b [ ; a] 2 ?
k
,
b
k
a: 2
2. The proof is similar.
3 The Characterization Theorems In this section we show that the Ginsberg-Fitting characterization of distributive billatices apply to IBLs in general, with the same construction. We assume, again, that is a xed IBL. B
3.1 Notation LB = x x f
j
RB = x x
t ?g
f
j
t ?g
3.2 Proposition 1. LB = x x f
j
k
2. The relations other.
t ;
RB = x x
g
t
f
and
k
j
k
f
g
on LB are identical while on RB they are inverse to each
Proof 1. RB = [f; ] = [f ?
2:2
t
;f
?
] = [ ; f] = x x
?
k
?
k
f
j
k
f . The proof for LB is similar. g
2. Immediate from Part 1, 2.9 and 2.3.
3.3 Theorem If B is an IBL then there are bounded lattices L; R such that These L and R are unique up to isomorphism.
7
B
is isomorphic to L R.
Proof We rst prove uniqueness. So assume is isomorphic to , where and are bounded lattices. Now, is obviously isomorphic to the sublattice (x; 0 ) x ; of (where here is that of ). But (x; 0 ) x is exactly LL R , which is isomorphic to L (since is isomorphic to ). Hence, is isomorphic to L ; (where here is that of ) and so is unique up to isomophism. Similarly, should be equivalent to RB ; , and so is also unique. To prove existence, we use the two condidates that are naturally suggested by the proof of uniqueness (and are the ones used also in [Gi88] and [Fi90] for distributive bilattices): B
L R
L
hf
t
L R
f
L R
B
t
j
R
R
j
R
2 Lg t i
L R
2 Lg
B
L
h
B
h
L
B
t i
R
t i
Let
L
B
R
B
= LB ; = RB ; h
t i
h
t i
(= LB ; (= RB ; h
k i
h
k i
; by 3:2) ; by 3:2) :
De ne g : B L R by g(x) = (x ; x ). We show that g is an isomorphism of on B B. That g is one-one is immediate from 2.8(1). To show that g is onto, let (a; b) L R . We show that g(a b) = (a; b). In other words, we show that if a b then, (i) (a b) = a; (ii) (a b) = b. For (i) note that since a a b, a (a b) by 2.4. On the other hand, since a b, a = a a a b. Since here also a it follows that a (a b) . Hence a = (a b) . The proof of (ii) is similar. Next we need to show that a b g(a) g(b) and a b g(a) g(b). We show the second quivalence (the proof of the rst is similar). So assume a b. Then and a b and a b . By 3.2 this means that a LB b . Hence g(a) = (a ; a ) (b ; b ) = g(b). Conversely, a RB b assume g(a) g(b) and so a b ,a b . This immediately entails that a = (a ) (a ) (b ) (b ) = b. 2 Other characterizations are easy consequences of 3.3. !
L
R
B
_?
B
^?
B
1
2
B
B
t ? t
k
t
_ ?
t
_ ?
_ ?
t
t ?
t
^ ?
,
t
t
k
,
_ ?
k
k
_ ? k
^ ?
_ ?
^ ? k
^ ?
_ ?
k
2:8
_?
^ ?
_ ? k
^?
k
_?
_ ?
^ ?
_ ?
^?
k
^ ? k
_ ?
_ ?
^ ?
^?
2:8
3.4 Proposition A structure B = hB; ; i is a complete IBL i there exist two complete lattices L; R such that B is isomorphic to L R. t
k
Most of the details of the proof are exactly as in [Fi90]. The proof that g is surjective is the only true innovation. Of course, what makes the reproducing of that proof possible is the demonstration in the previous section that the relevant facts obtain also for interlaced bilattices. 1
8
Proof The \if" part is straightforward. For the \only if" we note that if is complete then by an obvious generalization of 2.3, so are B and of the proof of 3.3. 2 B
L
RB
3.5 Proposition ([Gi88],[Fi90]) A structure B = hB; ; i is a (complete) distributive bilattice i there exist (complete) distributive bounded lattices L; R such that B is isomorphic to L R. t
k
Proof Again the \if" part is easy, while the \only if" part follows from the proof of 3.3 and the fact that if is a distributive bilattice Then ; are necessarily distributive lattices as well. B
L B RB
2
3.6 Proposition An IBL B is distributive i ^ and _ (or and ) are distributive over each other.
Proof Since B and lattice so are L
are de ned in terms of alone (or alone), if ; is a distributive 2 B and B . Hence so is also B B , which is isomorphic to .
RB L
t
R
k
L
R
hB t i
B
Note This means that for IBLs, the 12 conditions in the de nition of a distributive bilattice can be replaced by just two. We now establish a converse to Theorem 1.5, part (2).
3.7 Proposition Suppose B is a (complete, distributive) IBL with negation. Then there exist a (complete, distributive) bounded lattice L, so that hB; i is equivalent to L L, equipped with Ginsberg's negation (i.e. (x; y) = (y; x)).
9
Proof The proof is practically identical to that in [Fi94] for the distributive case: The function g de ned in the proof of 3.3 (g(x) = (x ; x )) is an isomorphism of on B B . On the other and the function x: x is easily seen to be an isomorphism between the lattices B and B . Hence the function h de ned by h(x; y ) = (x; y ) is an isomorphism between B B and B B . It follows that f = h g is an isomorphism between and B B . It remains only to show that f preserves also the negation operator. Now from the de nitions, f (x) = (x ; (x )) = (x ; x ) (see note (2) after De nition 1.3). Hence f ( x) = ( x ; x ) = ( x ; x ) = f (x) (according to Ginsberg's de nition of in B B ). 2 _?
^ ?
B
L
R
L
L
R
R
L
L
_ ?
L
^ ?
_ ?
_?
_ ?
_?
B
L
L
_ ?
_?
L
4 Applications In this section we give some examples of the power of the characterization theorem 3.3, which allows us to reduce claims about bilattices to simple calculations.
4.1 De nition 1. An element a of an IBL is called distributive if each equation of the form
x (y z) = (x y) (x z) 1
(were ;
2
1
2
1
; ; ; ) obtains in case x = a or y = a or z = a.
1 2 2 f_ ^ g
2. Distributive elements of a lattice are de ned similarly.
4.2 Theorem ; ; t and f are all distributive (in any IBL
> ?
B
).
Proof It is enough, by 3.3, to check it for IBLs of the form . Obviously, the equation obtains there i the corresponding distribution equations obtain for each component separately. Since ; ; t; f are all de ned in terms of the extreme elements of and , and since such elements of a lattice are always distributive (trivial), the claim follows. 2 L R
> ?
L
10
B
An example Suppose
1
= , _
2
= , x = (x ; x ), y = t = (1 ; 0 ), z = (z ; z ),
1
2
L
1
R
2
(x ; x ) ((1 ; 0 ) (z ; z )) = (x ; x ) (1 z ;0 z ) = (x ; x ) (z ; 0 ) = (x z ; x 0 ) = (x z ; 0 ) ((x ; x ) (1 ; 0 )) ((x ; x ) (z ; z )) = (1 ; 0 ) (x z ; x z ) = (x z ; 0 ) : 1
2
_
2
_
L
R
1
2
1
1
_
L
uL
1
1 tL
1
L
R
1
2
_
1
2
L
R
R
1
uR
2
1
2 uR
1 tL
2
1 tL
R
2 uR
1
_
1
R
1
R
1 tL
2
1
R
Note In [Fi90], Fitting pointed out that in his proof of the analogue of 3.3 for distributive bilattices, only instances involving of the distributive laws are used. Hence the theorem is valid for every IBL in which is distributive (and similarly for t; f; ). It follows that an alternative proof of 3.3 can be achieved if we prove 4.2 directly. This is possible, but the proof is longer. One of the remarkable properties of all known nite IBLs is that they can be represented by a two-dimensional graph, in which one axis represents the relation, while the other the relation. Moreover, the same edge on the graph between points A and B means that A and B are immediate successors according to both relations (although it is possible that A is an immediate -successor of B while B is an immediate -successor of A). In [Av95] we have shown that all nite IBLs have in fact such a graphic representation. The key for this was the next result. With the help of 3.3, we can obtain a new easy proof of it. ?
?
>
t
k
t
k
4.3 Proposition Let a
Bilattices were introduced and applied by Ginsberg and Fitting for a diversity of applications, such as truth maintenance systems, default inferences and logic programming. In this paper we investigate the structure and properties of a particularly important class of bilattices called interlaced bilattices (introduced by Fitting). The main results are that every interlaced bilattice is isomorphic to the Ginsberg-Fitting product of two bounded lattices and that the variety of interlaced bilattices is equivalent to the variety of bounded lattices with two distinguishable distributive elements which are complements of each other . This implies that interlaced bilattices can be characterized using a nite set of equations. Our results generalize to interlaced bilattices results of Ginsberg, Fitting and Jonsson for distributive bilattices.
Introduction The notion of a bilattice was introduced by Ginsberg in [Gi88] as a general framework for a diversity of applications, such as truth maintenance systems, default inferences and others. The notion was further investigated and applied for logic programming by Fitting ([Fi89], [Fi90], [Fi91], [Fi94]). The main idea behind bilattices is to use structures in which there are two partial order relations, having di erent interpretations. There should, of course, be a connection between the two relations. Ginsberg uses for this an operation of negation which is order preserving w.r.t. one, an involution w.r.t. to the other. Another connection he considered is distributive laws (12 altogether). This was generalized by Fitting who introduced the notion of an interlaced bilattice, in which all the basic bilattice operations are order preserving w.r.t. both partial orders. The structure of distributive bilattices is well understood since Ginsberg and Fitting proved a characterization theorem for them, to the e ect that each such bilattice is isomorphic to a certain product of two distributive bounded lattices. Fitting has further observed that if we apply that construction to any two bounded lattices (not necessarily distributive) 1
we get an interlaced bilattice. The converse, however, was not established, and the structure of interlaced bilattices has not, so far, been well understood. In this paper we thoroughly investigate the structure of interlaced bilatttices (with and without negation), and their important properties. Our main result is a generalization to interlaced bilattices of the Ginsberg-Fitting characterization of distributive bilattices (i.e. the converse of Fitting's observation). Other central results are characterizations using equational bases and the equivalence of the variety of interlaced bilattices with the variety of bounded lattices with two complementary, distributive elements. (This generalizes to interlaced bilattice results of Jonsson in [Jo94] for distributive bilattices.)
Note In order to make the presentation complete and self-contained, we repeat, together with the appropriate references, some short proofs which have already appeared elsewhere.
1 General Background
1.1 De nition [Fi90]
An interlaced bilattice (IBL) is a structure = B; ; ; ; ; ; ; t; f; ; B
h
t k ^ _
> ?i
such that
1. B; ; ; ; t; f is a bounded lattice (with the order relation, and the meet and join operations, and t, f the maximal and minimal elements, respectively). h
t ^ _
i
2. B; ; ; ; ; h
k > ?i
t
^
_
is also a bounded lattice.
3. Each of the four operations ; ; ; is order preserving with respect to both (e.g. if a b then a c b c). _ ^
k
k
^
k
t
and
^
Note In the original de nitions of Ginsberg and Fitting B; and b; are required to be complete lattices. We shall call below IBLs with this property complete. h
t i
h
k i
1.2 De nition [Gi88] An IBL is called distributive if all the twelve possible distributive laws concerning ; ; and hold.
^ _
2
Note Fitting has observed (see, e.g. [Fi90]) that the distributive laws imply condition (3) of 1.1. For example if a b then b = a b, and so c b = c (a b) = (c a) (c b) by the distributive laws. Hence c a c b in this case. t
_
t
_
_
1.3 De nition A unary operation on an IBL is called negation if it is order-preserving with respect to and an involution w.r.t. . In other words, is negation if
B
k
t
a=a a b b a b a
(i) (ii) (iii)
t
)
k
)
a b:
t
t
Notes 1. Ginsberg's original de nition of a bilattice in [Gi88] is: a structure which satis es the rst two conditions in de nition 1 (+ completeness) and has a negation. 2. Obviously, if is a negation then (a b) = a a b, (a b) = a b, t = f , f = t,
^
b, (a b) = a
_
>
_
= , >
?
= .
b, (a b) =
^
?
The main method of constructing an IBL is described in the following de nition. It was essentially introduced by Ginsberg in [Gi88], and further generalized by Fitting in [Fi90].
1.4 De nition Let = L; ; ; ; 1 ; 0 and = R; ; ; ; 1 ; 0 be two bounded lattices. Their product, , is the structure L R; ; ; ; ; ; ; t; f; ; de ned as follows: L
h
L tL uL
Li
L
R
L R
(i) (ii) (iii) (iv) (v) (vi) (vii)
h
h
R tR uR
Ri
R
t k ^ _
> ?i
(a ; b ) (a ; b ) a a and b b (a ; b ) (a ; b ) a a and b b (a ; b ) (a ; b ) = (a a ; b b ) (a ; b ) (a ; b ) = (a a ; b b ) (a ; b ) (a ; b ) = (a a ; b b ) (a ; b ) (a ; b ) = (a a ; b b ) t = 1 ;0 f = 0 ;1 = 1 ;1 1
1
k
2
2
,
1 L
2
1 R
2
1
1
t
2
2
,
1 L
2
1 R
2
1
1
_
2
2
1 tL
2
1 uR
2
1
1
^
2
2
1 uL
2
1 tR
2
1
1
2
2
1 tL
2
1 tR
2
1
1
2
2
1 uL
2
1 uR
2
Ri
>
h
L
Ri
h
L
3
h
L
Ri
?
= 0 ;0 h
L
Ri
1.5 Theorem 1. [Fi90,91], [Gi88] L R is an IBL. Moreover, if both L and R are distributive lattices then so is L R, and if both L and R are complete lattices then so is L R. 2. [Gi88] If L is a bounded lattice then the operation , de ned by (x; y) = (y; x), is a negation on L L.
The proof of this theorem is straightforward. For distributive bilattices, a converse to Theorem 1.5 was proved by Ginsberg and Fitting (in [Gi88], [Fi90]): if is a distributive bilattice then there exist distributive bounded lattices and such that is isomorphic to . These and are unique (up to isomorphism). Ginsberg-Fitting's result will easily follow from a generalization we prove below for every IBL. B
L
R
B
L R
L
R
2 Basic Properties of IBLs In this section is a xed interlaced bilattice. B
2.1 De nition [a; b] = x B a [a; b] = x B a
(1) (2)
t
f
2
j
t
f
2
j
k
k
x b x b t
k
g g
2.2 Proposition 1. If a
t
b then [a; b] = [a b; a b] .
2. If a
k
b then [a; b] = [a b; a b] .
t
^
k
k
_
t
Proof We show the rst formula as an example. Assume rst that y [a; b] , so a y b. Since is interlaced, a (a b) y (a b) b (a b). Hence a b y (a b) a b and so y (a b) = a b. It follows that a b y. Similarly y a b. Hence y [a b; a b] . For the converse, assume that a b and a b y a b. Then a (a b) a y a (a b). But, if a b then a = a a a b and so a (a b) = a. Similarly, a (a b) = a. It follows that a a y a and so a y = a and a y. Similarly, y b and so y [a; b] . 2 2
B
t
t
k
^
t
^
k
2
t
k
k
t
^
^
k
t
4
t
k
k
t
t
t
2
t
^
^
^
t
t
k
k
2.3 Corollary [a; b] and [a; b] are closed under ; ; and . Moreover, in case a b then [a; b] is an interlaced bilattice, with the same order relations and bilattice operations as (but with a; b; a b; a b taking the roles of f; t; ; respectively). Similarly, if a b then [a; b] is an IBL with the same order relations and bilattice operations as . t
^ _
k
t
t
B
? >
k
k
B
2.4 Corollary If a
b
t
b then b f a a b and a
^ ? t
t
k
_?
k
b f and a t b a .
^ > t
t
k
b
k
a t. Similarly, if a
t
b then a [f; b] and b [a; t] .
k
b then
_ >
Proof The rst part follows from 2.2 and the fact that if a Similar considerations apply for the second part.
2
2
t
t
2.5 Corollary [Fi90] t f= ; =f ;
(i) (ii)
t f=
?
?^>
?_>
>
=t
Proof Since B = [f; t] , we have by 2.2 that B = [f t; f t] . But also B = [ ; ] . Hence, the two equations in (i). The proof of (ii) is similar.
t
? >
k
k
2.6 Proposition x x
(i) (ii)
^? ^>
=x f ; =x f ;
x x
_? _>
=x t; =x t:
Proof We show the rst equality. The proof of the rest is similar. Since x x we have by 2.4 that x f x . To show the converse it is enough to show that x x and x f (since is the meet operation of ). The rst inequality follows from x, since x x x x = x. The second inequality follows from f , since f x x f = f. 2 ^ ? t
^ ? k
^ ? k
? k
? k
k
? k
? k
)
^?
)
^ ? k
5
^ ? k
^
k
^
Notes 1. For distributive bilattices 2.6 was shown in [Jo94]. 2. If we substitute, e.g. for x in the rst equality we get another proof of Hence 2.5 is a corollary of 2.6. >
>^ ?
= f.
2.7 Proposition If x If x If x If x
(1) (2) (3) (4)
k
k
t
t
b then x = (x b then x = (x b then x = (x b then x = (x
^ ^
b) b) b) b)
_ ^
(x (x (x (x
_ _
b) b) b) b)
Proof Again we prove only (1). Assume x b. Then x x x b and so x x b. Similarly, x x b. By combining these two inequalities we get x (x b) (x b). On the other hand x [x b; x b] , and so , by 2.2, x [(x b) (x b); (x b) (x b)] . Hence, x (x b) (x b), and the quality follows. 2 k
^
k
k
2
^
^
k
^
_
^
^
k
_
_
2
_
k
k
^
^
_
_
2.8 Corollary (1) (2) (3) (4)
x = (x ) (x ) = (x f ) (x t) ; x = (x ) (x ) = (x f ) (x t) ; x = (x f ) (x f ) = (x ) (x ) ; x = (x t) (x t) = (x ) (x ) : ^?
_?
^>
_>
_
^
^?
_ ?
_
^
^>
_ >
Proof Immediate from 2.7 and 2.6.
Note For distributive bilattices 2.8 is an immediate corollary of 2.5 and 2.6 (an so { of 2.6) We end this section with a result which was shown for distributive bilattices in [Jo94]. 6
t
2.9 Proposition (1) (2)
k k
= =
t
t
i f = ? i t = > i f = > i t = ?
Proof 1. Obviously, if = then f = and t = . For the converse, assume, e.g. that f = . Then, for all a; b: k
b
t
a
,
t
b [f; a] 2
?
b [ ; a]
,
t
>
2 ?
t
2:2 ,
?
b [
2 ?
a;
?
a]
k
,
b [ ; a] 2 ?
k
,
b
k
a: 2
2. The proof is similar.
3 The Characterization Theorems In this section we show that the Ginsberg-Fitting characterization of distributive billatices apply to IBLs in general, with the same construction. We assume, again, that is a xed IBL. B
3.1 Notation LB = x x f
j
RB = x x
t ?g
f
j
t ?g
3.2 Proposition 1. LB = x x f
j
k
2. The relations other.
t ;
RB = x x
g
t
f
and
k
j
k
f
g
on LB are identical while on RB they are inverse to each
Proof 1. RB = [f; ] = [f ?
2:2
t
;f
?
] = [ ; f] = x x
?
k
?
k
f
j
k
f . The proof for LB is similar. g
2. Immediate from Part 1, 2.9 and 2.3.
3.3 Theorem If B is an IBL then there are bounded lattices L; R such that These L and R are unique up to isomorphism.
7
B
is isomorphic to L R.
Proof We rst prove uniqueness. So assume is isomorphic to , where and are bounded lattices. Now, is obviously isomorphic to the sublattice (x; 0 ) x ; of (where here is that of ). But (x; 0 ) x is exactly LL R , which is isomorphic to L (since is isomorphic to ). Hence, is isomorphic to L ; (where here is that of ) and so is unique up to isomophism. Similarly, should be equivalent to RB ; , and so is also unique. To prove existence, we use the two condidates that are naturally suggested by the proof of uniqueness (and are the ones used also in [Gi88] and [Fi90] for distributive bilattices): B
L R
L
hf
t
L R
f
L R
B
t
j
R
R
j
R
2 Lg t i
L R
2 Lg
B
L
h
B
h
L
B
t i
R
t i
Let
L
B
R
B
= LB ; = RB ; h
t i
h
t i
(= LB ; (= RB ; h
k i
h
k i
; by 3:2) ; by 3:2) :
De ne g : B L R by g(x) = (x ; x ). We show that g is an isomorphism of on B B. That g is one-one is immediate from 2.8(1). To show that g is onto, let (a; b) L R . We show that g(a b) = (a; b). In other words, we show that if a b then, (i) (a b) = a; (ii) (a b) = b. For (i) note that since a a b, a (a b) by 2.4. On the other hand, since a b, a = a a a b. Since here also a it follows that a (a b) . Hence a = (a b) . The proof of (ii) is similar. Next we need to show that a b g(a) g(b) and a b g(a) g(b). We show the second quivalence (the proof of the rst is similar). So assume a b. Then and a b and a b . By 3.2 this means that a LB b . Hence g(a) = (a ; a ) (b ; b ) = g(b). Conversely, a RB b assume g(a) g(b) and so a b ,a b . This immediately entails that a = (a ) (a ) (b ) (b ) = b. 2 Other characterizations are easy consequences of 3.3. !
L
R
B
_?
B
^?
B
1
2
B
B
t ? t
k
t
_ ?
t
_ ?
_ ?
t
t ?
t
^ ?
,
t
t
k
,
_ ?
k
k
_ ? k
^ ?
_ ?
^ ? k
^ ?
_ ?
k
2:8
_?
^ ?
_ ? k
^?
k
_?
_ ?
^ ?
_ ?
^?
k
^ ? k
_ ?
_ ?
^ ?
^?
2:8
3.4 Proposition A structure B = hB; ; i is a complete IBL i there exist two complete lattices L; R such that B is isomorphic to L R. t
k
Most of the details of the proof are exactly as in [Fi90]. The proof that g is surjective is the only true innovation. Of course, what makes the reproducing of that proof possible is the demonstration in the previous section that the relevant facts obtain also for interlaced bilattices. 1
8
Proof The \if" part is straightforward. For the \only if" we note that if is complete then by an obvious generalization of 2.3, so are B and of the proof of 3.3. 2 B
L
RB
3.5 Proposition ([Gi88],[Fi90]) A structure B = hB; ; i is a (complete) distributive bilattice i there exist (complete) distributive bounded lattices L; R such that B is isomorphic to L R. t
k
Proof Again the \if" part is easy, while the \only if" part follows from the proof of 3.3 and the fact that if is a distributive bilattice Then ; are necessarily distributive lattices as well. B
L B RB
2
3.6 Proposition An IBL B is distributive i ^ and _ (or and ) are distributive over each other.
Proof Since B and lattice so are L
are de ned in terms of alone (or alone), if ; is a distributive 2 B and B . Hence so is also B B , which is isomorphic to .
RB L
t
R
k
L
R
hB t i
B
Note This means that for IBLs, the 12 conditions in the de nition of a distributive bilattice can be replaced by just two. We now establish a converse to Theorem 1.5, part (2).
3.7 Proposition Suppose B is a (complete, distributive) IBL with negation. Then there exist a (complete, distributive) bounded lattice L, so that hB; i is equivalent to L L, equipped with Ginsberg's negation (i.e. (x; y) = (y; x)).
9
Proof The proof is practically identical to that in [Fi94] for the distributive case: The function g de ned in the proof of 3.3 (g(x) = (x ; x )) is an isomorphism of on B B . On the other and the function x: x is easily seen to be an isomorphism between the lattices B and B . Hence the function h de ned by h(x; y ) = (x; y ) is an isomorphism between B B and B B . It follows that f = h g is an isomorphism between and B B . It remains only to show that f preserves also the negation operator. Now from the de nitions, f (x) = (x ; (x )) = (x ; x ) (see note (2) after De nition 1.3). Hence f ( x) = ( x ; x ) = ( x ; x ) = f (x) (according to Ginsberg's de nition of in B B ). 2 _?
^ ?
B
L
R
L
L
R
R
L
L
_ ?
L
^ ?
_ ?
_?
_ ?
_?
B
L
L
_ ?
_?
L
4 Applications In this section we give some examples of the power of the characterization theorem 3.3, which allows us to reduce claims about bilattices to simple calculations.
4.1 De nition 1. An element a of an IBL is called distributive if each equation of the form
x (y z) = (x y) (x z) 1
(were ;
2
1
2
1
; ; ; ) obtains in case x = a or y = a or z = a.
1 2 2 f_ ^ g
2. Distributive elements of a lattice are de ned similarly.
4.2 Theorem ; ; t and f are all distributive (in any IBL
> ?
B
).
Proof It is enough, by 3.3, to check it for IBLs of the form . Obviously, the equation obtains there i the corresponding distribution equations obtain for each component separately. Since ; ; t; f are all de ned in terms of the extreme elements of and , and since such elements of a lattice are always distributive (trivial), the claim follows. 2 L R
> ?
L
10
B
An example Suppose
1
= , _
2
= , x = (x ; x ), y = t = (1 ; 0 ), z = (z ; z ),
1
2
L
1
R
2
(x ; x ) ((1 ; 0 ) (z ; z )) = (x ; x ) (1 z ;0 z ) = (x ; x ) (z ; 0 ) = (x z ; x 0 ) = (x z ; 0 ) ((x ; x ) (1 ; 0 )) ((x ; x ) (z ; z )) = (1 ; 0 ) (x z ; x z ) = (x z ; 0 ) : 1
2
_
2
_
L
R
1
2
1
1
_
L
uL
1
1 tL
1
L
R
1
2
_
1
2
L
R
R
1
uR
2
1
2 uR
1 tL
2
1 tL
R
2 uR
1
_
1
R
1
R
1 tL
2
1
R
Note In [Fi90], Fitting pointed out that in his proof of the analogue of 3.3 for distributive bilattices, only instances involving of the distributive laws are used. Hence the theorem is valid for every IBL in which is distributive (and similarly for t; f; ). It follows that an alternative proof of 3.3 can be achieved if we prove 4.2 directly. This is possible, but the proof is longer. One of the remarkable properties of all known nite IBLs is that they can be represented by a two-dimensional graph, in which one axis represents the relation, while the other the relation. Moreover, the same edge on the graph between points A and B means that A and B are immediate successors according to both relations (although it is possible that A is an immediate -successor of B while B is an immediate -successor of A). In [Av95] we have shown that all nite IBLs have in fact such a graphic representation. The key for this was the next result. With the help of 3.3, we can obtain a new easy proof of it. ?
?
>
t
k
t
k
4.3 Proposition Let a
