Dense Poly

8/12/2019 Dense Poly

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DENSIFICATION VIA POLYNOMIALS, LANGUAGES, AND FRAMES

NIKOLAOS GALATOS AND ROSTISLAV HORCIK

Abstract. It is known that every countable totally-ordered set can be embedded into a count-able dense one. We extend this result to totally ordered commutative monoids and to totally-ordered commutative residuated lattices (the latter result fails in the absence of commutativity).The latter has applications to density elimination of semilinear substructural logics. In partic-ular we obtain as a corollary a purely algebraic proof of the standard completeness of uninormlogic; the advantage over the known proof-theoretic proof and the semantical proof is that it isextremely short and transparent and all details can be verified easily using standard algebraicconstructions. As another corollary we obtain a modified form of amalgamation.

1. Introduction

1.1. Presentation style and modularity of the paper. One of the benefits of working in areasthat touch on different research subjects is that the main ideas of certain results can be adopted andpresented so as to be of interest to a range of readers. We therefore opt for a modular presentation,as we do not want to assume or impose knowledge of other fields to the reader. At the same time, forthe sake of interested readers, we include comments that illustrate the connections, in the contextof our special setting, between different research areas such as polynomial rings in classical algebra,syntactic congruences in language theory and semigroup theory, Dedekind-MacNeille completionsin order-theory, syntactic arguments for density eliminations in proof theory, as well as the theoryof nuclei and of residuated frames in residuated lattices and substructural logics. In particular,we give a very accessible and economical presentation of each result using the tools from the

corresponding area in order to avoid mentioning any general theory, and only in later sectionswe explain our design choices (for definitions that seem to be ad hoc) and the connections withrelated constructions using different tools, in order to make the paper useful for future researchattempts in related topics. A side goal of the paper is to demonstrate that methods and ideasfrom these distinct areas are actually very much related.

1.2. Results and layout. A feature of all of our constructions is that they produce a densestructure in steps by inserting (at least) one new element in a given gap in the original totally-ordered structure. A gap is simply a pair of elements g < h, such that no other element isbetween them. Each of the next three sections has two parts. The first part presents a proofof the corresponding construction in a spartan fashion, with no discussion of the choices of thedefinitions or any motivation. The second part has a detailed discussion and justification of the

definition by connecting to a more general theory. The final section aims at connecting thesethree sections and discusses the similarities and differences of the methods, essentially giving toour results case-study status in a more general project of linking different areas.

In detail, we prove that every totally-ordered commutative monoid can be embedded into adense one, by considering the natural syntactic congruence given by languages that separate thepoints of the given monoid from each other and from the additional point used in the densification.Readers with background in language theory or semigroup theory and interested in densificationof totally-ordered monoids should start by reading Section 2.

Then we show that also every totally-ordered commutative residuated lattice can be embeddedinto a dense one. The dense totally-ordered monoid given above may not include the residualsof the new elements, hence the need of a modification of the construction. For this we use linearpolynomials, where the indeterminate Xessentially plays the role of the element filling the givengap. Readers with background in classical algebra and ring theory should start their reading from

1


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2 NIKOLAOS GALATOS AND ROSTISLAV HORCIK

Section 3. As an aside we show that a modification of our construction leads to the standardapproach of adding square roots to (totally ordered) abelian groups.

An alternative approach which actually produces an order-complete residuated lattice is byusing the tool of residuated frames. By analyzing the proof using linear polynomials, we can

extract a very simplified version of the construction given in [1]; see Section 4.An application of densification to logic, and in particular to the standard completeness of certain

semilinear logics, is discussed in Section 1.4. A second application of our results is discussed inSection 5 and provides a variant of the amalgamation property for totally-ordered structures.

En route to proving the above results, we introduce novel constructions. In particular, wepresent a construction of a residuated lattice based on the polynomial semiring over a residuatedlattice. Also, we elaborate on the connections between modules over idempotent semirings andresiduated lattices. In particular, we explain how we can obtain residuated lattices as images ofmodule morphisms. Both of these constructions work in a more algebraic setting and have morealgebraic flavor than those employing residuated frames, as they do not force order-completenessof the resulting algebra.

1.3. Connections to existing results. Part of our motivation for considering densification of

totally-ordered algebras comes from substructural logic. More specifically, uninorm logic UL is thesemilinear extension of full Lambek calculus with exchange FLe [6], i.e., it is a logic complete withrespect to the class of all FLe-chains (namely, totally-ordered bounded commutative residuatedlattices) ; see [9]. In [9, 3] it is shown that the logic UL is standard complete, i.e., it is complete withrespect to the class of all bounded FLe-chains whose universe is the real unit interval [0, 1]. Theproof is based on a proof-theoretical elimination of the density rule and the inductive argumentsinvolve many cases, thus making it hard to verify them all.

A semantical proof inspired by this proof-theoretic argument is given in [1]. The semanticalproof simplifies the verification, but still involves a lengthy case analysis for checking the com-munication rule in order to establish the linearity of the resulting structure; the communicationrule directly corresponds to the proof-theoretic rendering of (pre)linearity. We provide a purelyalgebraic and also very short proof of densification (and thus also standard completeness). At

the same time, and more importantly, we provide an explanation of why the construction reallyworks, using basic tools from classical algebra. In particular, we believe that our explanationsjustify some of the ad hoc design choices in [3] and [1]. Furthermore, our approach provides asimpler methodology which can be used for checking densifiability of other logics while avoiding aconvoluted proof-theoretic case analysis or verification of the communication rule. Actually, thesimplified tools provided in this paper have already allowed us to prove densifiability for a largeclass of subvarieties axiomatized by equations in the signature of multiplication and join; as theinclusion of these results would make the current paper too lengthy and also distract from itsexplanatory nature, we present them in a separate forthcoming paper.

1.4. The one-step densification approach. In order to densify a countable totally orderedalgebra A, we first present a construction that given a gap g < h in A, namely a pair g < h ofelements ofA for which there is no element ofA between them, produces a new totally ordered

algebra Ainto which A embeds and which contains an element strictly between the images in

Aofg andh, thusfilling, or resolving, the gap; we call Aa one-step densificationorone-step linear

extension ofA. Of course this process could result in A having new gaps, even ifA had a singlegap.

However, having established the above, given a countable totally-ordered algebra A, we considerall countable totally-ordered algebras that have A as a subalgebra and we order them by thesubalgebra relation. By an easy application of Zorns Lemma to this poset we obtain the existenceof a maximal superalgebra B of A, which has to be dense by maximality (otherwise, if it has agap, an isomorphic copy ofB has B as a proper subalgebra).

Unfortunately, B need not be countable. Nevertheless, using the downward Lowenheim-Skolemtheorem (see [8, Cor. 3.1.4]), there is a countable elementary substructure C of B containing A.Since the property of being dense is a first-order property, C has to be dense as well. As C iscountable and dense its order reduct (C,

) is isomorphic to the rationals. It is well known that


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DENSIFICATION VIA POLYNOMIALS, LANGUAGES, AND FRAMES 3

the Dedekind-MacNeille completion of (C, ) is isomorphic to the interval [0, 1]. Moreover, in caseA, and thusC, is a totally-ordered commutative monoid or residuated lattice, we can define on theDedekind-MacNeille completion of (C, ) the same algebraic structure (see e.g. [6]), so it becomesa totally-ordered commutative monoid or residuated lattice based on [0, 1]. In the case of totally

ordered commutative residuated lattices (or commutative residuated chains), these algebras areknown as standardand our result implies that they generate (even via quasivariety generation)the variety of all semilinear commutative residuated lattices, namely the variety generated by allcommutative residuated chains.

2. Totally ordered monoids and syntactic congruences

Before presenting the construction of one-step densification we recall the definition of a com-mutative totally ordered monoid. A partially ordered monoid (or pomonoid) is a structure(M, , 1, ) where (M, , 1) is a monoid, is partial order on M and a b implies cad cbdfor all a,b,c,d M; it is called commutative if the monoid reduct is commutative. If the orderis total/linear, then M is said to be a totally ordered monoid (or tomonoid). Also, we define

a={mM :ma}.

2.1. The construction. Now given a commutative tomonoid A with a gap g < h we will con-struct a one-step linear extension of it, namely a commutative tomonoid A such that A is em-beddable into A via an embedding : A A. Moreover, there is an element p A such that(g)< p < (h).

The first and naive way to create a one-step densification is of course to add a single newelement p to the algebra A. The operations of the new algebra based on A {p} should extendthose ofA, as we require an embedding. In the new algebra, by order preservation and gph,we have agapah, for all aA. Therefore, a natural way to define multiplication by p is toset a p = ah (or similarly a g = ag ), for a A; of course we would have to exclude a = 1, aswe surely want 1

p = p. However, this causes complications with associativity. In particular, if

ab= 1 for some a, bA, then p = 1 p= (ab) p= a(b p) =a(bh) = (ab)h= h, a contradiction.So we have to proceed differently.

First we define a monoid extension ofAby the new element p and then look for a congruencewhich does not collapse the elements ofA {p} and is large enough to produce a totally-orderedquotient. We explain in Section 2.2.2 why it suffices to consider the following small extension. LetA Ap be the commutative monoid, where Ap ={ap|aA}(formally speaking we write ap for(a, p) and Ap for A {p}) and multiplication is given by stipulating that for all a, bA:

a b= ab a (bp) = (ab)p, (ap) (bp) = (abh)p.

Note that the order on A can be naturally extended to Ap by setting apbp iffab. ThenA Ap can be ordered as a disjoint union of posets (see Figure 1). Further, we can define a totalorder

on the subset A

{p

}of A

Ap, by simply stipulating that

extends the order of A

by the conditions g p h. In that sense A Ap contains an ordered part, namely the chain(A {p}, ); see Figure 1.

For every a, b A we define the set a b ={c A| ac b}. The following lemma statessome simple properties of such sets.

Lemma 2.1. Leta, b, cA. Then the following hold:(1) ab is downward closed.(2) 1b ={cA|cb}=b.(3) bc impliesabac.(4) bc impliescaba.(5) The collection{a b| a, b A} of subsets of A forms a chain under inclusion, as

downsets of a chain.


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gp

h

A Ap

g

h

p

1

g

(h

g)p

Lg

g

h

p

1

g

(1

1)p

Lp

g

h

p

1

h

(h

h)p

Lh

Figure 1. (Left) The monoid extension AAp together with the chain (A{p}, ) depicted by the thick line; (Right) The subsets Lg, Lp and Lh depictedby the thick lines.

We define the following subsets ofA Apin an ad hoc way and in Section 2.2.1 we justify thedefinition:

La= (1a) (ha)p for aA,Lp= (1g) (11)p,

where (bc)p={dpAp|dbc} for b, cA. The subsets Lg, Lp and Lh are depicted inFigure 1 (note that hg 11hh).

Given LA Ap, which we will call a language ofA Ap, and uA Ap, we define thequotient ofL by u as follows:

u1L={xA Ap|uxL}.We first show that there are only two types of sets of the form u1Lz, for uA Ap.

Lemma 2.2. LetuA Ap andzA {p}. Thenu1

Lz is of the formb1La= (ba) (bha)p or b1Lp= (bg) (b1)p

for somea, bA.Proof. Leta, bA. Then we have

b1La ={xA Ap|bxLa}={cA|bc1a} {cpAp|bcp(ha)p}={cA|bca} {cpAp|bcha}= (ba) (bha)p,

(bp)1La ={cA|bcp(ha)p} {cpAp|bchp(ha)p}= (bha) (bh2 a)p= (bh)1La,

b1Lp={cA|bc1g} {cpAp|bcp(11)p}= (bg) (b1)p,

(bp)1

Lp={cA|bcp(11)p} {cpAp|bchp(11)p}= (b1) (bh1)p=b1L1.

Lemma 2.3. For allzA {p} anduA Ap, we have thatu1Lz is a downset for.Proof. By Lemma 2.2, it suffices to prove that for all a, b A the sets b1La and b1Lp aredownsets for. To that goal letx, yA {p} with xy . Downset closure is trivially true forx = y, so we assume that x y for the rest of the proof. Note that the following implicationshold and also their hypotheses exhaust all cases:

ifx, yA then xy, ifx= p then hy,

ify= p then x

g.


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Fromyb1Lawe will derive xb1La. Ifx, yA thenyba. Sinceba is downwardclosed, we have x (b a) b1La. If x = p then h y and y b a, so bh by a.Thus 1 bh a, i.e., p (bh a)p b1La. Finally, if y = p then 1 bh a. Thusbx

bg

bh

a and x

(b

a)

b1La.

Similarly from yb1Lp we will obtain xb1Lp. Ifx, yA we can use the same argumentas above in order to show that xb1Lp. Ifx = p then hy and y bg , so bhby g .Thus b 1 (otherwise we would have h bh g < h). Consequently, 1 b 1, i.e.,p (b 1)p b1Lp. Finally, if y = p then 1 b 1. Thus b 1 which implies bx g.Consequently, xbgb1Lp.

For any pomonoid M, the syntactic preorderL (introduced in [13], see also [11]) given by alanguageLM is defined as follows: for x, yM,

xLy iff uM(yu1L = xu1L).The symmetrizationL =L (L)1 is the syntactic congruencecorresponding to the set L(also known as Leibniz congruence in Abstract Algebraic Logic; see e.g. [4]). It is easy to seethat the syntactic congruence

L is the largest monoid congruence which saturatesthe setL (i.e.,

aL and aLb impliesbL). Consequently,L separates the elements ofL from those in thecomplementLc ofL, that is ifa L and b Lc then aL b; here complements are computedrelative to M. Then the quotient monoid M/L, endowed with the orderinggiven by

[x]L[y]L iff xLy ,becomes a pomonoid; here [x]L denotes the congruence class of x with respect toL. Givenan indexed system of languages{Li| i I}, the intersection =

iILi is again a monoid

congruence and the quotient M/ forms a pomonoid where the order is given by

[x][y] iff xLi y for all iI .Here we consider the collection{Lz | z A {p}} of languages and the congruence =

zA{p}Lz .

Lemma 2.4. (AAp)/ is a pomonoid in which A is embeddable via a [a]. In addition,[g]


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c 1 bh a. By the assumption there is y b a such that y c g (i.e., by a andg < hcy). Let xc1, i.e., cx1. Then bhxbcyxbya, i.e., xbha. Theorem 2.6. (A Ap)/ is a tomonoid in which A is embeddable viaa [a]. In addition,[g]


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DENSIFICATION VIA POLYNOMIALS, LANGUAGES, AND FRAMES 7aA La would collapse congruence classes [c](p,h) and [d](p,h) for some c, d A such that

c < d. However, this is impossible since cLc d. Hence (p, h) =

aA La .This way of viewing as (p, h) Lp offers another explanation of how the linear extension

(A

Ap)/ was constructed. Namely, the principal congruence (p, h) has almost all desired

properties. It preserves the orderon A {p}and it separates all elements in A and also g fromp. Nevertheless, it collapses p and h. That is why we consider a finer congruence, namely theintersection (p, h)Lp . This step is reasonable because the syntactic congruenceLp separatesp from h and preserves the orderon A {p}.2.2.2. Small monoid extension. The choice of the monoid extension A Apseems ad hoc. A morereasonable candidate for the monoid extension is the free extension Ap, whereAp ={apn :aA, n N} under the multiplication given byapn bpm = (ab)pn+m. It turns out that such a largeextension is not necessary because if we would choose the free monoid extension Ap and followthe same strategy as above, the resulting congruence would contain the congruence generatedbyp2 =ph. Consequently, we can start immediately with Ap/=A Ap without changing thefinal result.

To see that is contained in, consider the same strategy as before looking for the smallest sets

Lz Ap such that they separate all element from A {p} and satisfy (1). Using the analogousarguments replacing A Ap with Ap, we would end up with the following sets:

La =

nN

(hn a)pn for aA,

Lp= (1g)

nN

(hn 1)pn+1.

By the definition of, it suffices to prove that Lz for every z A {p}. We clearly have[p2]Lz [ph]Lz (i.e., p2 Lz ph) since p h. Conversely, we have to show phLz p2, i.e.,p2 u1Lz implies ph u1Lz for all u Ap. Suppose thatp2 u1Lz. Thenup2 Lz forsome u = apk, a A and k N. Consequently, a is either in hk+2 b, if z = b A, or inhk+1 1, ifz = p. In the first case ahhk+1 b. Thus uph = ahpk+1 (hk+1 b)pk+1 Lb.In the second case ahh

k

1. Thus uph = ahpk+1

(hk

1)pk+1

Lp.3. Densification of commutative residuated chains, via linear polynomials

We first describe the construction in an economical and relatively ad hoc way so as to give avery short proof. In the next section we explain why these design choices are natural.

3.1. The construction. A commutative residuated lattice-ordered semigroup is a structure A=(A, , , , ) such that

(A, , ) is a lattice, (A, ) is a commutative semigroup and the residuation condition holds: x yzyx z, for all x, y,zA.

If there is an element 1A such that 1x= x for allxA then we callA acommutative residuatedlattice. If, in addition, the lattice order is total, A is said to be a commutative residuated chain. Itfollows easily from the residuation condition that the multiplication distributes over finite joins(i.e., x(yz) = xyxz). In particular, it is monotone in both arguments. Similarly, we have(x y) z = (x z) (y z) andx (y z) = (x y) (x z). Consequently the residuumis antitone in the first argument and monotone in second one. For details on residuated structuressee [6].

Let A be a commutative residuated lattice and let A(X) be the set{abX : a, b A} oflinear polynomials over the{, } reduct of A in one indeterminate X. This can be ordered bythe cartesian product ordering (asA(X) is isomorphic to A A) and it becomes a lattice. Clearly,A AXis not closed under the usual product of polynomials. However, given an element hA,we define in an ad hoc way the product of two linear polynomials p = p0 p1Xand q= q0 q1Xby:

(p0

p1X)

h(q0

q1X) =p0q0

(p0q1

p1q0

p1q1h)X.


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We further define implicationh on A(X) by(p0p1X) h(q0 q1X) = ((p0 q0) (p1 q1)) ((p0 q1) (p1h q1))X.

Lemma 3.1. The setA(X) forms a commutative residuated lattice-ordered semigroupA(X)underthe above operations.

Proof. The operationh is clearly commutative and it is straightforward to check that it is asso-ciative. To show that A(X) is residuated we have

(p0p1X) h(q0 q1X)r0 r1Xp0q0 (p0q1p1q0p1q1h)Xr0 r1Xp0q0r0, p0q1r1, p1q0r1, p1q1hr1q0 q1X((p0 r0) (p1 r1)) ((p0 r1) (p1h r1))Xq0 q1X(p0p1X) h(r0 r1X).

As it is a lattice, it follows that it is a commutative residuated lattice-ordered semigroup.

For all aA, we consider in an ad hoc fashion the linear polynomialsa= a (h a)Xand a= (a g) (a 1)X.

We denote by Athe set of these two types of polynomials, for all aA, and we writea for aanda for a if a long expression takes the place ofa.

Lemma 3.2. A is a totally-ordered subset ofA(X).

Proof. Indeed, ifab, then a band ba. We now compare a and b.Ifabg , then ab g. Also, ba g, so b(h a)(h a)(a g)h g


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Pha= (p0p1X) h((a g) (a 1)X)= ((p0 (a g)) (p1 (a 1))) ((p0 (a 1)) (p1h (a 1)))X= ((p0a g) (p1a 1)) ((p0a 1) (p1ha 1))X= ((p0a g) (p0a 1)X) ((p1a 1) (p1ha 1)X)= p0a p1a 1

Sinceis a closure operator by Lemma 3.3 whose image AsatisfiesPh Q Afor allP A(X)andQ A, it follows thatis anucleusonA(X), namely a closure operator satisfying (x)(y)(xy), so A is a commutative residuated lattice-ordered semigroup; see e.g. [6, Section 3.4.11] fordetails.

Next we have to show that1 is a multiplicative unit. We have

1 a= (1 ha) =(a ((h a) a(h 1) h(h 1)(h a))X) =(a) = a,because a(h 1), h(h 1)(h a)h a. Similarly,

1 a= ((ag) ((a1) (ag)(h1) h(h1)(a1))X) =(a) = a.By Lemma 3.2 A is a chain, so it is a commutative residuated chain.We also have g


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there is a unique commutative residuated lattice A on the setA= A {, }specified by thefollowing conditions

A is a subalgebra ofA,

< a 0}, so Ap = Ap+ A =App A. Using properties of the action and lifting to sets elementwise we have

Ap(A {p}) (ApA) (Ap{p})

(ApA)

((App

A)

{p

})

(ApA) (App{p}) (A{p})(p(AA)) (p(A({p}{p}))) (A{p})

DefiningY PZ={y z :yY , zZ} and making the identifications p(AA)p(A P

A) pA h P A A and p(A({p}{p})) p(A P {1}) h P (A P {1}) A,stemming from the definition of in the frame, we conclude with the set A A{p}. In the abovewe wrote zz for{z} ={z}, and we liftedto sets of elements in an elementwise fashion.4.2.3. Adding square roots to an abelian group. To obtain the construction of Section 3.3, wemodify the multiplication of W = A Ap so that p2 = h. Then picking W as above does notyield an action in general, so we defineW =W (A {p}). However, it turns out that the basicclosed sets again fall under two categories ifg = 1 and h is its cover.

5. Mixed amalgamation

The one-step densification described in the previous sections can be also viewed as a form ofamalgamation. More precisely, given a commutative residuated chain A with a gap g < h andits one-step densification A, one can view A as an amalgam of A and the three element chain{g < p < h} over the two element chain{g < h} (see Figure 3).

Here we will show that the above result can be further generalized. In particular, any com-mutative residuated chain Bcan be amalgamated with any chain Cover any chain A to producea commutative residuated chain D. The analogous result also holds for commutative tomonoids.Note that usual amalgamation for commutative residuated chains neither follows from nor im-plies the above mixed amalgamation property. In particular, densification is not really related tothe usual amalgamation property as there is no a priori prescribed structure of the subalgebra

generated by the new element(s) in the densified algebra.We will make use of the following auxiliary constructions. Given a commutative residuatedchain A, one can define a residuated chain on the setU(A) of all upward closed subsets of A.Indeed, it is easy to see that the map :P(A) P(A) defined by (S) =S, whereS is theupset generated by S, is a nucleus on the residuated latticeP(A) = (P(A), , , , , {1}). ThusU(A) =P(A) is a residuated chain. Moreover, the map a {a}is an order-reversing injectionpreserving multiplication in A. If we apply this construction twice, we obtain a residuated chainU(U(A)) and an order-embedding : A U(U(A)) defined by (a) =Ua, whereUa={{a}}={U U(A)| a U}. In addition, it is a residuated lattice homomorphism. It clearly preservesmultiplication so it suffices to check that it preserves. We have

Ua Ub={V U(A)| UaV Ub}=

{V

U(A)

| U

Ua :

(U

V)

Ub

}={V U(A)| U Ua : b (U V)}={V U(A)|b ({a} V)}={V U(A)| vV :avb}={V U(A)| vV :va b}=Uab.

In fact, the above construction is just the canonical extension ofA (details on canonical extensionsfor residuated lattices can be found in [6, Chapter 6]).

Now we are ready to describe a construction which plugs a new element into any place within agiven commutative residuated chain A. LetGA be a downset, we will define an extension AGofA that inserts a new element just above all the elements ofG and below all elements in A \ G.IfG has a maximum g which is covered by h (i.e., g < h forms a gap) then AG is defined as theone-step densification ofA. OtherwiseG has no maximum element, or it has a maximum element


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A

B

C

D

fB

fC

gB

gC

g

h

gph

g

h

gph

A

A

Figure 3. (Left) The one-step densification as amalgamation; (Right) The V-formation and its completion.

but its complement has no minimum element (in other words G does not define a gap g < h).Then we define AG to beU(U(A). This construction creates new elements for each infinitary joinand infinitary meet, so AG will contain one or two new elements between G and its complement.In all of the above cases we have a construction which extends A by at least one element whichis above all elements in G and below all elements in A \ G. We will denote by the embedding ofA into AG and actually we replace the image of by the original elements ofA, thus converting into an inclusion map and Ainto a subalgebra ofAG.

Theorem 5.1. LetA, C be chains, letB be a commutative residuated chains and letfB :A BandfC :A C be order embeddings. (The tuple(A, B, fB, C , f C) is often called a V-formation.)Then there is a commutative residuated chainD, an order embeddinggC :CD and a residuatedlattice embeddinggB :BDsuch that the right diagram in Figure 3 commutes, i.e.,gBfB =gCfC.Proof. Without any loss of generality we identify the elements ofA with their images in CunderfC. So we assume thatACandfCis just the inclusion map. Using the axiom of choice, we canindex the elements ofC

\Aby an ordinal , i.e.,C

\A=

{c

C

\A

| <

}. We will construct

the algebra D starting from B and inserting the elements ofC\ Aby transfinite recursion.We define C0= A, D0= B and g0= fB. Further, we consider the chain{C|}ordered

by inclusion, where C =A {cC\ A| < }, i.e., C contains all elements from A plus theelements inC\A indexed by ordinals less than . The minimum of this chain is C0= A and themaximum is C= C.

Now we define algebrasD and the maps g : CD for ordinals 0< . If= +1 forsome ordinal . Then define D = (D)G for G =g[{c} C]. The mapg is an extensionofg mapping c to an element in (D)G which is above all elements in G and below all elementsin D\ G. Thus g is an order-embedding and g|C = g so the corresponding inner diagramcommutes in Figure 4.

Ifis a limit ordinal then we define D =


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A= C0 C1 C2

B= D0 D1 D2

g0 g1 g2

C=C

D= D

g

Figure 4. The proof of the mixed amalgamation.

nonstandard notation

to denote the union operation, but considered as the meet of the module.

Indeed, Xi Zj =

(XiZj ) and Y(XZ) = (X

Y)Z.

We also know that sinceP(W) is a residuated lattice, (P(W),) is also a module over(P(W),, , {}), where the action is.

Note that the map : (P(W),)(P(W),) is a module morphism. Indeed, we haveX Y N Z iff Y N XZ. Namely, Y X Z iff X Y Z iff Y (XZ). So,(XZ) = XZ. Also, ( Zi) =Zi . The image of this map is an (P(W), , , {})-submodule of (P(W),) and actually an intersection-closed structure inP(W), so it forms aresiduated lattice and is exactly the dual algebra W+ of the frame W.

6.2. From residuated frames to polynomials. Given a (commutative) residuated lattice A,we consider the residuated frame W= (W, W, N, , 1, ) we defined in Section 4.1. To show therelationship between this residuated frame and the residuated latticeA(X), we consider the subsetX =

{p

}ofW. By collecting all the monomialscpn for the same n together, we note that each

subsetC ofWcan be written as C=C0 C1X, where Cn={cA : cpn C}.Note that multiplication in the powersetP(W) is given by the formula

(C0 C1X)(D0 D1X) =C0D0 (C0D1 C1D0 C1D1{h})Xand join is defined as the componentwise union. The resulting structure is a semiring which issimilar to A(X) in the sense that we replaced the coefficient semiring A by (P(A), , , {1}) andXis not an abstract indeterminate this time but the set{p}.

We also note that in the case when A is complete, for every principal downset D A andarbitrary subset C A, we have that C D P(A) is also a principal downset; also theintersection of principal downsets is also principal. In other words the principal downsets form anuclear retraction of (P(A), , , , , {1}), where the nucleus is given by S (S).

We extend this function to the powerset ofW =A Ap, under the direct product ordering, bysettingC0 C1X C0 (C1)X. This defines also a nucleus now on the residuated latticeP(W) = (P(W), , , , , {1}), whose image is isomorphic to the semiring (actually residuatedlattice) A(X).

Since A(X) is a nuclear retraction of the residuated latticeP(W), it is also a homomorphicimage of the semiringP(W) and also aP(W)-submodule ofP(W). It turns out that the finaldual algebra W+ is a further submodule of it. Thus in case that A is complete the approach ofresiduated frames and the approach using polynomials coincide by simply taking X={p}.

6.3. From residuated frames to languages. Finally, we compare the approaches using resid-uated frames and syntactic congruences. Assume that we have a commutative residuated chainA with a gap g < h, is the pomonoid congruence on AAp defined in Section 2 and isthe nucleus induced by the residuated frame W defined in Section 4.1. We claim that x y iff(

{x

}) =(

{y

}) for all x, y

A

Ap.


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Indeed, first note that if A is residuated then the language b1La for a, b A is just thelanguage Lba. Further note that for a, b A we have{a} = La and{bp} = b1Lp. Thusb1La={b a}. Consequently, we have (cf. Lemma 2.2)

x aA La y iff xLa y for all aAiff xb1Layb1La for all a, bAiff x {b a} y {b a} for all a, bAiff x {a} y {a} for all aA.

Similarly, we have (cf. Lemma 2.2)

xLp y iff xu1Lpyu1Lp for all uA Apiff xb1Lpyb1Lp andxb1L1yb1L1 for all bAiff x {bp} y {bp} andx {b 1} y {b 1} for all bA.

Using the above facts, we finally obtain the following chain of equivalences:

x y iff x aA

Lz y andxLp y

iff x {a} y {a} and x {bp} y {bp} for all a, bAiff x {z} y {z} for all zA Apiff ({x}) =({y}).

6.4. Combinations with completions. Given a commutative residuated chain A with a gapg < h, the approach with residuated frames produces a complete residuated chain, the one withpolynomials produces a residuated chain (which may not be complete) and the approach withlanguages yields simply a tomonoid, and the three results sit one inside the other. However, thelast two sit as appropriate substructures of the first, and to be precise the first complete residuatedchain is the Dedekind-MacNeille completion of each of the other two. In that sense the approachby residuated frames subsumes the other two, as one can then take the residuated sublattice orsubtomonoid of the complete residuated chain and obtain the desired result.

The approaches via residuated frames and polynomials suppose that the original algebra A isa commutative residuated chain. Thus it is not clear whether we can use it in order to obtaina one-step densification of a commutative tomonoid. Nevertheless, every commutative tomonoidcan be embedded into a commutative residuated chain using a nuclear completion. Namely, onecan take the nuclear retraction ofP(A) generated by downsets{abA|a, bA}(cf. [2, 10]).

Thus any of the three approaches can be applied, in combination with completions at appro-priate places, to subsume the others. All these are of course not clear in the beginning as thetools and methods come from different disciplines and this is why we felt compelled to describein detail the three approaches and their connections hoping to initiate a partial unification of thethree subjects.

Acknowledgment

The work of the first author was supported by Simons Foundation 245806, FWF project STARTY544-N23. The work of the second author was partly supported by the grant P202/11/1632 ofthe Czech Science Foundation and partly by the long-term strategic development financing of theInstitute of Computer Science (RVO:67985807).

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Nombres, 9:124, 1956.

Department of Mathematics, University of Denver, 2360 S. Gaylord St., Denver, USA

E-mail address: [email protected]

Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vod arenskou vez 2,

Prague, Czech Republic

E-mail address: [email protected]

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