Unification and Projectivity in De Morgan and Kleene Algebras

0.31 MB·

by  Simone Bova

Unification and Projectivity
in De Morgan and Kleene Algebras
Simone Bova and Leonardo Cabrer
4
1
0 Abstract
2
Weprovideacompleteclassification ofsolvableinstancesoftheequa-

n tional unification problem over De Morgan and Kleene algebras with re-
a specttounificationtype. Thekeytoolisacombinatorialcharacterization
J
of finitely generated projective DeMorgan and Kleene algebras.
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1
1 Introduction

]
O
A De Morgan algebrais a boundeddistributive lattice with aninvolutionsatis-
L
fying De Morganlaws; that is, a unary operationsatisfying x=x′′ andx∧y =
.
h (x′∨y′)′. A Kleene algebra is a De Morgan algebra satisfying x∧x′ ≤ y∨y′.
t
a In [11], Kalman shows that the lattice of (nontrivial) varieties of De Morgan
m algebras is a three-element chain formed by Boolean algebras (Kleene algebras
[ satisfying x∧x′ = 0), Kleene algebras, and De Morgan algebras, and these

varietyarelocallyfinite; thatis,finite, finitelypresented,andfinitely generated
1
algebras coincide.
v
Inavarietyofalgebras,the symbolic(equational)unificationproblemisthe
6
7 problemofsolvingfinite systemsofequationsoverfreealgebras. Aninstanceof
5 the symbolic unification problem is a finite system of equations, and a solution
3 (aunifier)isanassignmentofthe variablesto termssuchthatthesystemholds
.
1 identically in the variety. The set of unifiers of a solvable instance supports
0 a natural order, and the instances are classified depending on the properties
4
of their maximal unifiers. In this paper, we provide a complete (first-order,
1
decidable)classificationofsolvableinstancesofthe unificationproblemoverDe
:
v Morgan and Kleene algebras with respect to their unification type.
i
X The keytooltowardsthe classificationis acombinatorial(first-order,decid-
able) characterization of finitely generated projective De Morgan and Kleene
r
a algebras, motivated by the nice theory of algebraic (equational) unification in-
troducedby Ghilardi[9]. In the algebraicunificationsetting,aninstance ofthe
unification problem is a finitely presented algebra in a certain variety, a uni-
fier is a homomorphismto a finitely presentedprojective algebrain the variety,
andunifiers supporta naturalorder that determines the unification type of the
instance, in such a way that it coincides with the unification type of its finite
presentation, viewed as an instance of the symbolic unification problem.
Even if projective Boolean algebras have been characterized in [3, 15], a
complete characterization of projective De Morgan and Kleene algebras lacks
in the literature. In this note, also motivated by an effective application of the
algebraic unification framework, we initiate the study of projective De Morgan
1
and Kleene algebras, and relying on finite duality theorems [7], we provide a
combinatorialcharacterizationof finitely generated projective algebras,and we
exploit it to classify all solvable instances of the equationalunification problem
over De Morgan and Kleene algebras with respect to their unification type;
in particular, we establish that De Morgan and Kleene algebras have nullary
equational unification type (and avoid the infinitary type).
Thepaperisorganizedasfollows. InSection2,wecollectfromtheliterature
the background on projective algebras, duality theory, and unification theory
necessary for the rest of the paper. For standard undefined notions and facts
in order theory, universal algebra, category theory, and unification theory, we
refer the reader to [8], [13], [12], and [2] respectively. In Section 3, we intro-
duce the characterization of finite projective De Morgan and Kleene algebras.
In Section 3, we introduce the characterization of finite projective De Morgan
and Kleene algebras (respectively, Theorem 11 and Theorem 12). In Section 4,
we provide complete classification with respect to unification type of all solv-
able instances of the equational unification problem over bounded distributive
lattices, Kleene algebras, and De Morgan algebras (respectively, Theorem 15,
Theorem 22, and Theorem 30). The distributive lattices case tightens previous
results by Ghilardi [9], and outlines the key ideas involved in the study of the
more demanding cases of Kleene and De Morgan algebras.
2 Preliminaries
Let P = (P,≤) be a preorder, that is, ≤ is reflexive and transitive. If x and
y are incomparable in P, we write x k y. Given X,Y ⊆ P, we write X ≤
Y iff x ≤ y for all x ∈ X and y ∈ Y; we freely omit brackets, writing for
instance x ≤ y,z instead of {x} ≤ {y,z}. If X ⊆ P, we denote by (X] and
[X) respectively the downset and upset in P generated by X, namely (X] =
{y ∈ P | y ≤x for some x∈X} and [X) = {y ∈ P | y ≥x for some x∈X}; if
X = {x} we freely write (x] and [x). If x,y ∈ P, we write [x,y] = {z ∈ P |
x ≤ z ≤ y}. A set X ⊆ P is directed if for all x,y ∈ X there exists z ∈ X
such that x,y ≤ z. We denote minimal elements in P by min(P) = {x ∈ P |
y ≤x implies x≤y for all y ∈P}. Similarly we denote maximalelements in P
by max(P). Let P=(P,≤) and Q=(Q,≤) be preorders. A map f: P →Q is
monotone if x≤y implies f(x)≤f(y).
2.1 Projective Algebras
LetV beavarietyofalgebrasandκbeanarbitrarycardinal. AnalgebraB∈V
is saidto have the universal mapping property for κ if there exists X ⊆B such
that |X| = κ and for every A ∈ V, and every map f: X → A there exists
a (unique) homomorphism g: B → A extending f (any x ∈ X is said a free
generator, and B is said freely generated by X). For every cardinal κ, there
exists a unique algebra with the universal mapping property freely generated
by a set of cardinality κ, called the free κ-generated algebra in V, and denoted
by FV(κ).
Since the varieties of De Morganand Kleene algebras,in symbols M and K
respectively, are generated by single finite algebras [11], they are locally finite,
that is, finitely generated and finite algebras coincide.
2
Example 1. By direct computation, FM(1) is the bounded distributive lattice
over {0,x∧x′,x,x′,x∨x′,1} shown in Figure 1.
1
x∨x′
x x′
x∧x′
0
Figure 1: FM(1).
Let V be a variety of algebras. An algebra A∈V is said to be projective if
foreverypairofalgebrasB,C∈V,everysurjectivehomomorphismf: B→C,
and every homomorphism h: A→C, there exists a homomorphism g: A→B
such that f ◦g =h.
We exploit the following characterizationof projective algebras [10].
Theorem 2. Let V be a variety, and let A∈V. Then, A is projective in V iff
A is a retract of a free algebra FV(κ) in V for some cardinal κ, that is, there
exist homomorphisms r: FV(κ)→A and f: A→FV(κ) such that r◦f =idA.
2.2 Finite Duality
We recall duality theorems for the categories of finite bounded distributive lat-
tices, D , finite De Morgan algebras, M , and finite Kleene algebras, K .
f f f
First,wepresentBirkhoffdualitybetweenfiniteboundeddistributivelattices
andfinite posets[6]. The categoryP offinite posets has finite posets (P,≤) as
f
objects, and monotone maps as morphisms.
Define the map J: D →P as follows: For every A in D , let
f f f
J(A)=({x|x join irreducible in A},≤),
where≤is the orderinheritedfromthe orderinA. Foreveryh: A→BinD ,
f
let J(h): J(B)→J(A), be the map defined by
J(h)(x)= {y |h(y)≥x},
for all x∈J(B). ^
Define the map D: P →D as follows: For every P=(P,≤)∈P , let
f f f
D(P)=({X ⊆P |(X]=X},∩,∪,∅,P).
3
For every f: P→Q in P , let D(f): D(Q)→D(P) be the map defined by
f
D(f)(X)=f−1(X)
for all X ∈D(Q).
Theorem 3 (Birkhoff, [6]). J and D are well defined contravariant functors.
Moreover, they determine a dual equivalence between D and P .
f f
BuildingonthedualityforboundeddistributivelatticesdevelopedbyPriest-
ley [14], in [7, Theorem 2.3 and Theorem 3.2] Cornish and Fowler present a
duality for De Morgan and Kleene algebras. We rely on Theorem 3, to restrict
such duality to finite objects.
Definition 4 (Finite Involutive Posets, PM and PK ). The category PM
f f f
of finite involutive posets is defined as follows:
Objects: Structures (P,≤,i), where (P,≤) is a finite poset, and i: P → P is
such that x≤y implies i(y)≤i(x) and i(i(x))=x.
Morphisms: Maps f: (P,≤,i) → (P′,≤′,i′) such that x ≤ y implies f(x) ≤′
f(y) and f(i(x))=i′(f(x)).
The category PK is the full subcategory of PM whose objects (P,≤,i) are
f f
such that i(x) is comparable to x for all x∈P.
The map JM: Mf →PMf is defined by: For every A=(A,∧,∨,′,0,1) in
M , let
f
JM(A)=(J(A,∧,∨,0,1),i),
where i(x) = (A \{a′ | a ∈ [x)}) for each x ∈ J(A,∧,∨,0,1). Moreover,
JM(h)=J(h) for every h: A→B in FM.
The map DVM: PMf →Mf is defined by: For every P=(P,≤,i)∈PMf,
DM(P)=A=(A,∧,∨,′,0,1)
where (A,∧,∨,0,1) = D(P,≤), and X′ = P \i(X). Moreover, DM(f) = D(f)
for every f: P→Q in PM .
f
We will denote JK: Kf → PKf and DK: PKf →Kf the restrictions of the
functors JM and DM to the categories Kf and PKf respectively.
Theorem 5 (Cornishand Fowler). JM and DM (respectively, JK and DK) are
welldefinedcontravariantfunctors. Moreover,theydetermineadualequivalence
between the categories M and PM (respectively, K and PK ) .
f f f f
Let D = (D,≤,i) ∈ PM be as in Figure 2. In light of Example 1,
f
JM(FM(1)) and D are isomorphic via the map x∧x′ 7→ 2, x 7→ 0, x′ 7→ 1,
17→3.
Let P = (P,≤,i) ∈ PM . By [7, Theorem 2.4], the product of n copies of
f
P in the category PM , denoted by
f
Pn =(Pn,≤n,in),
isthefinite posetoverPn withthe orderandtheinvolutiondefinedcoordinate-
wise, that is for all x = (x ,...,x ),y = (y ,...,y ) ∈ Pn, x ≤n y iff x ≤ y
1 n 1 n i i
for all i=1,...,n, and in(x)=(i(x ),...,i(x )).
1 n
4
3
0 1
2
Figure 2: JM(FM(1))≃D. Curved edges depict the map i: D →D.
Proposition 6. JM(FM(n))≃Dn.
Proof. FM(n)isthecoproductofncopiesofFM(1). Therefore,byTheorem5,
JM(FM(n)) ≃ JM(FM(1))n ≃ Dn, the product of n copies of JM(FM(1)).
The statement follows.
33
03 30 31 13
00 01 23 32 10 11
02 20 21 12
22
Figure 3: JM(FM(2))≃D2.
Let P = (P,≤,i) ∈ PM . By [1], subobjects of P are subsets X ⊆ P
f
with the inherited order such that X = i(X). By Theorem 5, subobjects of P
correspondexactly to quotients on DM(P). For each P in PMf, let Pk be the
largestsubobjectofPlyinginthesubcategoryPK ,thatis,P isthesubobject
f k
of P (possibly empty) such that each element x of P is comparable with i(x).
k
Therefore, DM(Pk) is the largest quotient of DM(P) lying in Kf.
Proposition 7. JK(FK(n))≃(Dn)k.
Proof. FK(n) is the largest quotient of FM(n) that is a Kleene algebra. By
Proposition 6, JM(FM(n)) ≃ Dn. Then by the mentioned correspondence be-
tweenquotients and subobjects under the duality [1], JK(FK(n))=JM(FK(n))
arises as the largest subobject of Dn lying in PK .
f
5
33
03 30 31 13
00 01 10 11
02 20 21 12
22
Figure 4: JK(FK(2))≃(D2)k.
2.3 Unification Theory
Let P=(P,≤) be a preorder. A µ-set for P is a subsetM ⊆P such that xky
for all x,y ∈ M such that x 6= y, and for every x ∈ P there exists y ∈ M such
that x≤y. It is easy to check that if P has a µ-set, then every µ-set of P has
the same cardinality.
We say that P has type:
nullary if P has no µ-sets (in symbols, type(P)=0);
infinitary if P has a µ-set of infinite cardinality (type(P)=∞);
finitary ifPhasafinite µ-setofcardinalitygreaterthan1(type(P)=ω);
unitary if P has a µ-set of cardinality 1 (type(P)=1).
We prepare for later use some easy consequences of the definitions.
Lemma 8. The set {0,ω,∞,0} carries a natural total order 1 ≤ ω ≤ ∞ ≤ 0.
If P is a preorder and Q⊆P be an upset of P and Q denotes the preorder with
universe Q and relation inherited from P, then type(Q)≤type(P).
Lemma 9. Let P = (P,≤) be a directed preorder. Then, type(P) = 0 or
type(P)=1.
ThealgebraicunificationtheorybyGhilardi[9]reducesthe traditionalsym-
bolic unification problem over an equational theory to the following:
Problem Unif(V).
Instance A finitely presented algebra A∈V.
Solution A homomorphismu: A→P, where P is a finitely presented projec-
tive algebra in V.
A solution to an instance A is called an (algebraic) unifier for A, and A is
called solvable in V if A has a solution.
6
LetA∈V be finitely presented,andfori=1,2letu : A→P be aunifier
i i
for A. Then, u is more general than u , in symbols, u ≤ u , if there exists
1 2 2 1
a homomorphism f: P → P such that f ◦u = u . For A solvable in V, let
1 2 1 2
UV(A) be the preorder induced by the generality relation over the unifiers for
A. WedefinethetypeofAasthetypeofthepreorderedsetUV(A),insymbols
typeV(A)=type(UV(A)).
We say that the variety V has type:
nullary if {type (A)|A solvable in V}∩{0}6=∅;
V
infinitary if ∞∈{type (A)|A solvable in V}⊆{∞,ω,1};
V
finitary if ω ∈{type (A)|A solvable in V}⊆{ω,1};
V
unitary if {type (A)|A solvable in V}={1}.
V
3 Finite Projective Algebras
We providefirst-orderdecidable characterizationsofthe finite involutiveposets
corresponding to finite projective De Morgan(Theorem 11) and Kleene (Theo-
rem 12) algebras.
Definition 10 ([5]). Let κ be a cardinal. A poset (P,≤) is κ-complete if,
whenever X ⊆ P is such that all Y ⊆ X with |Y| < κ have an upper bound,
then X exists in (P,≤).
TheoWrem 11 (De Morgan Projective). Let A ∈ Mf. Then A is projective in
M iff JM(A)=(P,≤,i)∈PMf satisfies the following:
(M ) (P,≤) is a nonempty lattice;
1
(M ) for all x∈P, if x≤i(x), then there exists y ∈P such that x≤y =i(y);
2
(M ) {x∈P |x≤i(x)} with inherited order is 3-complete.
3
Proof. There exists n ∈ N such that JM(A) = (P,≤,i) = P is a subobject of
JM(FM(n))=Dn =(Dn,≤,i)∈PMf byProposition6. Thatis,itispossible
to display P as a subset of Dn with inherited order and involution. Combining
this together with Theorem 2 and Theorem 5, A is projective iff there exists
an onto morphism r: Dn → P in PMf such that r◦r = r, that is r|P = idP,
where r| denotes the restriction of r to P. Therefore, it is sufficient to show
P
thatconditions (M )-(M ) arenecessaryandsufficient forthe existence ofsuch
1 3
a map r.
Below, Z ={z ∈Dn |z =i(z)}={0,1}n and Y =Z∩P.
(⇒) Letr: Dn →P be a morphismin PMf suchthat r|P =idP. We show
that P satisfies (M ), (M ), and (M ).
1 2 3
For (M ): In particular, r is a poset retractionofDn onto P. Since Dn is a
1
nonempty lattice, it follows straightforwardlythat P is a nonempty lattice.
For (M ): Let y ∈P be such that y ≤i(y). Then y ∈{2,0,1}n. Let z ∈Z
2
be such that for i = 1,...,n, if y ∈ {0,1} then z = y and z = 0 otherwise.
i i i i
Then y ≤z ≤i(y), and y =r(y)≤r(z)=r(i(z))=i(r(z)).
For(M ): Observethatr((Z])=(Y] ,because if x≤z forx∈Dn andz ∈
3 P
Z, then r(x) ≤ r(z) ∈Y. Then the restriction of r to (Z] is a poset retraction
7
of (Z] onto (Y] . Since (Z] = {2,0,1}n is 3-complete, by [5, Corollary 2.6]
P
r((Z])=(Y] is 3-complete.
P
(⇐) Assume that P satisfies (M ), (M ), and (M ). We show that there
1 2 3
exists a morphism r: Dn →P in PMf such that r|P =idP.
To define the retraction r: Dn → P, we first introduce the following nota-
tion. Forall x∈Dn, letL ={z ∈P |z ≤x} andU ={z ∈P |x≤z}. Since
x x
i is an order reversing involution,
i(L )=U , (1)
x i(x)
for each x∈Dn.
Ifx∈Z,thenthereexistsy ∈Y issuchthat L ≤y ≤ U . Infact,if
P x P x
z ,z ∈L , then z ,z ≤x≤i(z ),i(z ). By (M ), z ∨ z ≤i(z )∧ i(z )=
1 2 x 1 2 1 2 1 1 P 2 1 P 2
W V
i(z ∨ z ). Combining this with (M ), we have L ≤ i( L ). Now, by
1 P 2 3 P x P x
(M ), there exists y ∈Y suchthat L ≤y. Finally by (1) and the fact that
2 P x W W
x = i(x), L ≤ y = i(y) ≤ i( L ) = i(L ) = U = U , as
P x WP x P x P i(x) P x
desired.
W W V V V
For each x∈Z, we fix r(x)∈Y such that
L ≤r(x)≤ U . (2)
P x P x
If x ∈ Dn \ Z, then let mWbe the smallesVt number in {1,...,n} such that
x ∈{2,3}. We define,
m
L , if x =2;
r(x)= P x m (3)
(WP Ux, if xm =3.
The map r: Dn → P is well deVfined. Also, r(x) = x for all x ∈ P because
x∈U ∩L .
x x
Claim 1: For each x∈Dn, r(i(x))=i(r(x)).
If x ∈ Z, then r(i(x)) = i(r(x)) holds because r(x) ∈ Y. Let x ∈ Dn\Z,
and m be the smallest number in {1,...,n} such that x ∈ {2,3}. If x = 2
m m
and (i(x)) =3, then r(x)= L and r(i(x))= U . Then by (1),
m P x P i(x)
r(i(x)W)= U = i(LV)
P i(x) P x
=i( L )
V P x V
=i(r(x)).
W
Thecasex =3and(i(x)) =2reducesto thepreviouscase,whichconcludes
m m
the proof of the claim.
Claim 2: r is monotone.
Let x,y ∈Dn such that x<y. Then L ⊆L , U ⊆U . Therefore,
x y y x
L ≤ L ≤ U , (4)
P x P y P y
and W W V
U ≤ U . (5)
P x P y
If x ∈ Z, observe that y ∈ {0V,1,3}n \{V0,1}n. By (2), (3) and (5), r(x) ≤
U ≤ U =r(y). A similar argument proves that r(x)≤r(y) if y ∈Z.
P x P y
V V
8
If x,y ∈ Dn \Z. If r(x) = L , then r(x) ≤ r(y) by (4) and (3). If
P x
r(x)= U , thenby (3), letting mbe the smallestnumber in{1,...,n}such
P x W
that x = 3 since x ≤ y it follows that y ∈ {0,1,3} for every k < m, and
m k
V
y = 3. Again by (3), r(y) = U . Therefore, r(x) ≤ r(y) by (5), which
m P y
concludes the proof of the claim.
V
By Claim 1 and Claim 2, r: Dn → P is the required retraction, so that A
is a retract of FM(n) in Mf, that is, it is projective in Mf.
Since (P,≤) is a finite lattice by (M ), condition (M ) reduces to the fol-
1 3
lowing first-order statement: x∨y∨z ≤ i(x∨y ∨z), for all x,y,z such that
x∨y ≤i(x∨y), x∨z ≤i(x∨z), and y∨z ≤i(y∨z).
Theorem 12 (Kleene Projective). Let A ∈ K . Then A is projective in K iff
f
JK(A) = (P,≤,i) ∈ PKf satisfies conditions (M2), (M3) in Theorem 11 and
the conditions:
(K ) {x∈P |x≤i(x)} with inherited order is a nonempty meet semilattice;
1
(K ) every x,y ∈ P such that x,y ≤ i(y),i(x) have a common upper bound
2
z ∈P such that z ≤i(z).
Proof. There exists n ∈ N such that JK(A) = (P,≤,i) = P is a subobject of
JK(FK(n))=(Dn)k =((Dn)k,≤,i)∈PKf byProposition7. CombiningTheo-
rem2andTheorem5,Aisprojectiveiffthereexistsamorphismr: (Dn) →P
k
in PKf such that r|P = idP. Therefore, it is sufficient to show that conditions
(K ), (K ), (M ), and (M ) are necessary and sufficient for the existence of
1 2 2 3
such a map r.
Below, Z ={z ∈(Dn) |z =i(z)}={0,1}n and Y =Z∩P.
k
(⇒) Let r: (Dn)k →P be a morphism in PKf such that r|P =idP.
The proof that P satisfies (M ) and (M ) follows by the same argument
2 3
used in the proof of Theorem 11.
For (K ): First observe that r(Z) = Y and r((Z]) = (Y] . Then the
1 P
restriction of r to (Z] is a poset retraction of (Z] onto (Y] . Since (Z] is a
P
nonemptymeetsemilattice,(Y] isanonemptymeetsemilattice[5,Lemma2.4].
P
For(K ): Letx,y ∈P, be suchthat x,y ≤ i(x),i(y). Then there does not
2 P
exist i ∈ {1,...,n} such that x = 0 and y = 1 (otherwise, x k i(y)), which
i i P
proves that x∨Dn y ∈ {2,0,1}n = (Z] ⊆ Dkn. Then z = r(x∨Dn y) ∈ (Y]P.
Therefore, x,y ≤z ≤i(z), as desired.
(⇐)LetPbeasubsetof(Dn) withinheritedorderandinvolutionsatisfying
k
(K ), (K ),(M )and(M ). We define amorphismr: (Dn) →PinPK such
1 2 2 3 k f
that r|P =idP.
SinceP∈PK ,by(M )wehavethatP =(Z]∪[Z)=(Z]∪i((Z]). Moreover,
f 2
sinceZ isanantichain,(Z]∩[Z)=Z. Forallx∈(Z],letL ={z ∈P |z ≤x}.
x
ObservethatL ⊆(Y] by(M ). Also,ifv,w ∈L ,thenv,w ≤x≤i(v),i(w),
x P 2 x
and combining (K ) and (K ), we obtain that v∨ w∈(Y] for all v,w ∈L .
1 2 P P x
Therefore, L ∈(Y] by (M ).
P x P 3
For all x∈Z, we define r(x)∈Y such that,
W
L ≤r(x), (6)
P x
W
9
whoseexistenceisensuredbycondition(M ). Andforallx∈(Z]\Z,wedefine,
2
r(x)= L , (7)
P x
r(i(x))=i(r(x)). (8)
W
The map r: (Dn) → P is well defined. We prove that r is the desired
k
retraction. Clearly, if x ∈ P, then r(x) = x. By definition, r commutes with
i. We check monotonicity. Let x,y ∈ (Dn) be such that x < y. If x,y ∈ (Z],
k
then L ⊆L , then
x y
r(x)= L ≤ L ≤r(y),
P x P y
where the last inequality alwayWs holds byW(6) and (7). If x ∈ (Z] and y ∈ [Z),
then there exists z ∈ Z such that x ≤ z ≤ y, so that i(y) ≤ z. Then
r(x),r(i(y)) ≤ r(z) by the previous case, but r(i(y)) = i(r(y)) by commuta-
tivity of r, so that r(z) = r(i(z)) = i(r(z)) ≤ r(y) by the properties of i and
commutativity of r. If x,y ∈ [Z), then i(x),i(y) ∈ (Z] and i(y) ≤ i(x), then
r(i(y))≤r(i(x)), then i(r(y))≤i(r(x)), and so r(x)≤r(y).
Observe that the conditions (K ), (K ) are first-order conditions on the set
1 2
{x∈P |x≤i(x)}.
4 Classification of Unification Problems
Weobtainacomplete,decidable,first-orderclassificationofunificationproblems
overboundeddistributive lattices(Theorem15), Kleenealgebras(Theorem22)
and De Morgan (Theorem 30) algebras with respect to unification type. In
particular, we establish that unification over the varieties of De Morgan and
Kleene algebras is nullary.
Forthe sakeofpresentation,weintroduce the followingnotion. An alphabet
Σ is asetofletters. Aword overΣ is afinite sequenceoflettersin Σ. Aformal
language over Σ is a subset of words over Σ.
4.1 Distributive Lattices
We classify all solvable instances of the unification problem over bounded dis-
tributivelatticeswithrespecttotheirunificationtype(Theorem15),thustight-
ening the nullarity result by Ghilardi in [9]. This case study prepares the tech-
nically more involved cases of Kleene and De Morgan algebras.
In[4],BalbesandHorncharacterizeprojectiveboundeddistributivelattices.
In the finite case, the characterizationstates that a finite bounded distributive
lattice L∈D is projective iff the finite poset J(L)∈P is a nonempty lattice.
f f
Thus, combining the algebraic unification theory developed by Ghilardi [9] and
the finite duality by Birkhoff (Theorem 3), a unification problem over bounded
distributive lattices reduces to the following combinatorial question:
Problem Unif(DL).
Instance A finite poset P.
Solution A monotone map u: L→P, where L is a finite nonempty lattice.
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