Commutative Residuated Lattices

James B. Hart, Lori Rafter, and Constantine Tsinakis

September 21, 2000

Abstract

A commutative residuated lattice, is an ordered algebraic struc-ture L = (L, ·,∧,∨,→, e), where (L, ·, e) is a commutative monoid,(L,∧,∨) is a lattice, and the operation → satisfies the equiva-lences

a · b ≤ c ⇐⇒ a ≤ b → c ⇐⇒ b ≤ c → a

for a, b, c ∈ L. The class of all commutative residuated lattices,denoted by CRL, is a finitely based variety of algebras. Histor-ically speaking, our study draws primary inspiration from thework of M. Ward and R. P. Dilworth appearing in a series ofimportant papers [9] [10], [19], [20], [21] and [22]. In the en-suing decades special examples of commutative, residuated lat-tices have received considerable attention, but we believe thatthis is the first time that a comprehensive theory on the struc-ture of residuated lattices has been presented from the viewpointof universal algebra. In particular, we show that CRL is an ”idealvariety” in the sense that its congruences correspond to order-convex subalgebras. As a consequence of the general theory, wepresent an equational basis for the subvariety CRLc generated byall commutative, residuated chains. We conclude the paper byproving that the congruence lattice of each member of CRLc is an

2000 Mathematics Subject Classification Primary 06B05; Secondary 06A15, 06B10,06B20, 06D20, 06F05, 06F07, 06F20

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algebraic, distributive lattice whose meet-prime elements forma root-system (dual tree). This result, together with the mainresults in [12] and [18], will be used in a future publication to an-alyze the structure of finite members of CRLc. A comprehensivestudy of, not necessarily commutative, residuated lattices will bepresented in [4].

1 Preliminaries.

We begin with some notation. Let (P,≤) be a poset. We will use > and ⊥to denote the greatest and least elements, respectively, of P when such exist.Let X ⊆ P . We will let ↑ X denote the upperset generated by X in P ; thatis

↑ X = {p ∈ P : x ≤ p, ∃ x ∈ X}.If X = {x} is a singleton, we will use the symbol ↑ x in place of the morecumbersome ↑ {x}. Uppersets of the form ↑ x will be called principal. Thelowerset, ↓ X, generated by X is defined dually.

A commutative binary operation · : P ×P → P on a poset (P,≤) is saidto be residuated provided there exists a binary operation →: P × P → Psuch that

a · b ≤ c ⇐⇒ b ≤ a → c ⇐⇒ a ≤ b → c.

In this event, we will say that (P,≤) is a residuated poset under the operation·, and refer to the operation → as a residual of ·. Note that for all a, b ∈ P ,

a → b = max {p ∈ P : a · p ≤ b}.

The definition above serves as a natural extension of the notion of anadjoint pair in the sense that, for all a ∈ P , the assignment x 7→ a → xserves as the right adjoint for the map y 7→ a · y. For an extensive discussionof adjunctions, see Gierz et al. [11].

As a consequence of the general theory of adjunctions, the binary op-eration · preserves all existing joins in both coordinates, and the residualpreserves all existing meets in its second coordinate. Residuals of the oper-ation · enjoy an additional property which will often prove useful. For anyfixed a ∈ P , observe that

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y ≤ x → a ⇐⇒ x · y ≤ a ⇐⇒ x ≤ y → a.

Hence, the operation → is dually self adjoint. Therefore, it is anti-isotone inits first component, converting all existing joins to meets (in P ).

By a commutative residuated lattice, we shall mean an ordered algebraicstructure of the form L = (L, ·,∧,∨,→, e), where

• (L, ·, e) is a commutative monoid,

• (L,∧,∨) is a lattice, and

• the operation → serves as the residual for the monoid multiplicationunder the lattice ordering.

We denote the class of all such objects by CRL.We wish to advise the reader that we do not require the monoid identity to

be the greatest element of the lattice. This constitutes a significant departurefrom the aforementioned work of Ward and Dilworth. See also Blyth [6] andMcCarthy [16].

As an example, note that every Brouwerian algebra is a member of theclass CRL where → in this case denotes the classical Brouwerian implication.It is worth noting that we are lax here with regard to the similarity type.Thus, we view a Brouwerian algebra as a member of the subvariety of CRLsatisfying the additional identity x ·y = x∧y. Likewise, every abelian lattice-ordered group (`-group) may be viewed as a member of the subvariety of CRLsatisfying the additional identity x · (a → e) = e. In this case, x → e = x−1,and, more generally, x → y = x−1 · y. (For information on Brouwerianalgebras, see Balbes and Dwinger [2] and Kohler [14]; for information on`-groups, see Anderson and Feil [1].)

It is a routine matter to verify that a binary operation → is the residualof a commutative, associative operation · on a lattice (L,∧,∨) if and only ifthe following identities hold:

1. x · (y ∨ z) = (x · y) ∨ (x · z)

2. x → (y ∧ z) = (x → y) ∧ (x → z)

3. x · (x → y) ∨ y = y

4. x → (x · y) ∧ y = y

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Thus, the class CRL is a variety.We have mentioned two important subvarieties of CRL, namely the sub-

variety of Brouwerian algebras and the subvariety of abelian `-groups. Sincethe Brouwerian implication residuates the lattice meet operation, it is clearthat any Brouwerian algebra is distributive. Likewise, it is well-known thatthe lattice reduct of any `-group is also distributive (see for example Ander-son and Feil [1]). It is certainly not necessary, however, for the lattice reductof a residuated lattice to be distributive. There are a number of ways toconstruct nondistributive residuated lattices; one of the simplest ways wasdevised by Peter Jipsen and is presented below.

Example 1.1

Let (L,∧,∨,⊥,>) be any bounded lattice containing an atom e. Definea binary operation · on L as follows.

x · y =

> if x, y 6∈ {⊥, e}x if y = e

y if x = e

⊥ if x = ⊥ or y = ⊥

It is easy to verify that the operation · is a multiplication on L withmultiplicative identity e . It is also easy to verify that

x → y =

> if x = ⊥ or y = >e if ⊥ < x ≤ y < >⊥ if x 6≤ y

serves as the residual of the multiplication.

Among other things, the construction in Example 1.1 can be used toresiduate any finite lattice, producing a distinct residutated multiplicationfor each atom.

We conclude this section by presenting a result that collects some basicproperties of the arrow operation. Its simple proof is left to he reader.

Lemma 1.2 Let L be a member of the variety CRL. For all a, b, c, d ∈ L, thefollowing statements are true:

1. e → a = a, and e ≤ a → a

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2. a · (a → b) ≤ b

3. a → (b → c) = (a · b) → c

4. (a → b) → (c → d) = c → [(a → b) → d]

5. (a → e) · (b → e) ≤ (a · b) → e.

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2 Convex Subalgebras

The primary aim of this section is to establish that the congruence latticeof any member of CRL is isomorphic to the lattice of its convex subalge-bras. In addition, we provide a canonical description of the elements of theconvex subalgebra generated by an arbitrary subset. We begin with someterminology.

We will say that a subset C of a poset (P,≤) is order-convex (or simplyconvex) in P if, whenever a, b ∈ C, then ↑ a∩ ↓ b ⊆ C.

As usual, we will refer to a subset H of a commutative, residuated lattice Las being a subalgebra of L provided H is closed with respect to the operationsdefined on L. We will let SubC(L) denote the set of all convex subalgebrasof L, partially ordered by set-inclusion. It is easy to see that the intersectionof any family of convex subalgebras of L is again a convex subalgebra of L;hence, SubC(L) is a complete lattice in which meet is set-intersection.

In the work to follow, we will let Con(L) denote the lattice of congruencerelations for a member L of the variety CRL. Since the congruence latticeof any lattice, in particular that of the lattice-reduct of L, is an algebraicdistributive lattice, the same is true for Con(L). For all θ ∈ Con(L), set

Hθ = {a ∈ L : (a, e) ∈ θ}.

For any convex subalgebra H of L, set

θH = {(a, b) ∈ L × L : a · h ≤ b and b · h ≤ a ∃ h ∈ H}= {(a, b) ∈ L × L : (a → b) ∧ e ∈ H and (b → a) ∧ e ∈ H}

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The first characterization of θH is the same as the one introduced by Mc-Carthy in [16] for lattice-ordered semigroups. We shall use this characteriza-tion of θH and leave it to the reader to prove that the second characterizationis equivalent. Our next goal will be to prove that the assignments θ 7→ Hθ

and H 7→ θH establish an order isomorphism between Con(L) and SubC(L).

Lemma 2.1. Let L be a member of CRL. If θ ∈ Con(L), then Hθ is a convexsubalgebra of L.

Proof. We first prove convexity. Suppose that a, b ∈ Hθ, and suppose x ∈ Lis such that a ≤ x ≤ b. We must prove that (x, e) ∈ θ.

Since a ≤ x, we know a∨x = x. Thus, since (a, e), (x, x) ∈ θ, (x, x∨e) ∈ θ.Since x ≤ b, x = x∧ b. Thus, since (b, e) ∈ θ, we see that (x, (x∨ e)∧ e) ∈ θ.However, since e = e ∧ (e ∨ x), we are done.

We now prove that Hθ is a subalgebra of L. To this end, let a, b ∈ Hθ.Since (a, e), (b, e) ∈ θ, it follows at once that (a∨ b, e), (a∧ b, e), and (a · b, e)are all members of θ. Furthermore, since e → e = e by Lemma 1.2 (1), wesee that (a → b, e) ∈ θ as well.

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Lemma 2.2. Let L be a member of CRL. If H is a convex subalgebra of L,then θH is a congruence relation on L.

Proof. It is routine to prove that θH is an equivalence relation. To establishthe substitution properties, let (a, b), (c, d) ∈ θH . By definition, there existh, j ∈ H such that

• a · h ≤ b and b · h ≤ a,

• c · j ≤ d and d · h ≤ c.

Clearly, we may replace h and j by k = h ∧ j. Since the multiplication isisotone and preserves joins, it is easy to see that

• (a · c) · k2 ≤ b · d and (b · d) · k2 ≤ a · c and

• (a ∨ c) · k ≤ b ∨ d and (b ∨ d) · k ≤ a ∨ c.

Consequently, (a · c, b · d), (a ∨ c, b ∨ d) ∈ θH .To see that (a∧c, b∧d) ∈ θH , first observe that a ≤ k → b and c ≤ k → d.

Hence,

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a ∧ c ≤ (k → b) ∧ (k → d) = k → (b ∧ d).

Thus, we have (a ∧ c) · k ≤ (b ∧ d). The proof that (b ∧ d) · k ≤ (a ∧ c) issimilar.

It remains to prove that (a → c, b → d) ∈ θH . Since b · k ≤ a, we knowby Lemma 1.2 (2) that

(b · k) · (a → c) ≤ a · (a → c) ≤ c.

Therefore, (b · k2) · (a → c) ≤ k · c ≤ d, which implies

k2 · (a → c) ≤ b → d.

The other inequality follows in like manner.2

Theorem 2.3. If L is a member of CRL, then Con(L) is order isomorphicto SubC(L). The isomorphism is established via the assignments θ 7→ Hθ andH 7→ θH .

Proof. It is easy to see that the assignments are both isotone. It willtherefore suffice to prove that θHθ

= θ and HθH= H.

Let θ ∈ Con(L) and observe that

θHθ= {(x, y) ∈ L × L : x · h ≤ y and y · h ≤ x ,∃ h ∈ Hθ}.

If (a, b) ∈ θ, then ((a → b) ∧ e, e) ∈ θ and ((b → a) ∧ e, e) ∈ θ by Lemma 1.2(1). Thus, if we let

h = (a → b) ∧ (b → a) ∧ e,

then it is clear that h ∈ Hθ. Furthermore, we see by Lemma 1.2 (2) that

• a · h ≤ a · (a → b) ≤ b, and

• b · h ≤ b · (b → a) ≤ a.

Hence, we see that (a, b) ∈ θHθ; it follows that θ ⊆ θHθ

. On the other hand,suppose that (x, y) ∈ θHθ

. Then there exist h ∈ L such that

• (h, e) ∈ θ, and

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• x · h ≤ y and y · h ≤ x.

Consequently, we know ([x · h] ∨ y, x ∨ y) ∈ θ and ([y · h] ∨ x, x ∨ y) ∈ θ.Transitivity and symmetry therefore imply

([y · h] ∨ x, [x · h] ∨ y) ∈ θ.

Since x = [x · h]∨ x and y = [y · h]∨ y, we see that (x, y) ∈ θ. It follows thatθHθ

⊆ θ.Now, let H ∈ SubC(L) and observe that

HθH= {a ∈ L : (a, e) ∈ θH}.

Suppose a ∈ H. Then h = a∧ (a → e) ∈ H; hence, we know e ·h ≤ e ·a ≤ a.By Lemma 1.2 (2), we also know a · h ≤ a · (a → e) ≤ a. Hence, (a, e) ∈ θH ;and we see that H ⊆ HθH

. On the other hand, suppose that a ∈ HθH. Then

(a, e) ∈ θH , which implies that there exists h ∈ H such that

a · h ≤ e and e · h ≤ a.

It follows that h ≤ a ≤ h → e; hence, a ∈ H by convexity.2

We now turn attention to obtaining a simple internal characterization ofconvex subalgebras. Let L be a member of CRL and let H be any subalgebraof L. Borrowing terminology from lattice-ordered groups, we will call the set↓ e ∩ H the negative cone of H. We will denote the negative cone of H byH−. In symbols, we have

H− = {h ∈ H : h ≤ e}.

Corollary 2.4. If L is a member of CRL, then every convex subalgebra ofL is completely determined by its negative cone. More specifically, if H is aconvex subalgebra of L, then

H = {x ∈ L : h ≤ x ≤ h → e , ∃ h ∈ H−}.

Proof. Let H be a convex subalgebra of L. Theorem 2.3 implies

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H = HθH= {x ∈ L : (x, e) ∈ θH}= {x ∈ L : x · h ≤ e and h ≤ x ,∃ h ∈ H}= {x ∈ L : h ≤ x ≤ h → e , ∃ h ∈ H}= {x ∈ L : h ∧ e ≤ x ≤ (h ∧ e) → e , ∃ h ∈ H}= {x ∈ L : h′ ≤ x ≤ h′ → e , ∃ h′ ∈ H−}.

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Note that the negative cone of any residuated lattice obtained by Example1.1 contains only two elements. Hence by Corollary 2.4, any such algebra issimple in the sense that its congruence lattice contains exactly two members.

Let L be a member of CRL. In what follows, we will adopt a standardnotation from semigroups. For all n < ω, and for all a ∈ L, let

1. a0 = e, and

2. an = a · an−1 for all 0 < n < ω.

The following observations will be useful when dealing with members ofthe negative cone.

Lemma 2.5. Let L be a member of CRL. If a, b, c are members of thenegative cone of L, then the following are true:

1. a ≤ a → e,

2. (a ∨ b) · (a ∨ c) ≤ a ∨ (b · c),

3. If n < ω, then (x ∨ y)n ≤ x ∨ yn,

4. If m,n < ω, then (x ∨ y)mn ≤ xm ∨ yn.

Proof. Claims (1) and (2) are obvious; we prove Claim (3) by means ofinduction. Since a ≤ e, it is clear that Claim (3) holds when n = 0. Assumingthe claim holds for any n < ω, observe that

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(a ∨ b)n+1 = (a ∨ b) · (a ∨ b)n

≤ (a ∨ b) · (a ∨ bn)

= a2 ∨ (a · bn) ∨ (b · a) ∨ bn+1

≤ a ∨ bn+1.

Claim (4) follows from Claim (3). Indeed, we have

(x ∨ y)mn = ((x ∨ y)n)m ≤ (x ∨ yn)m ≤ yn ∨ xm.

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For any L in CRL and any S ⊆ L, we will let C[S] denote the smallestconvex subalgebra of L containing S. As is customary, we will call C[S] theconvex subalgebra generated by S and will let C[a] = C[{a}]. We will callC[a] the principal convex subalgebra of L generated by the singleton {a} andusually refer to it as being generated by the element a. We will say that aconvex subalgebra of L is finitely generated provided it is generated by somefinite subset of L.

Lemma 2.6 If L is a member of CRL and S ⊆ L, then C[S] = C[S ′], whereS ′ = {e ∧ a ∧ (a → e) : a ∈ S}. In particular, every convex subalgebra of Lis generated by a subset of its negative cone.

Proof. Let a ∈ L and b = e ∧ a ∧ (a → e). It will suffice to prove thatC[a] = C[b]. First, note that b is clearly a member of the negative cone ofC[a]; hence, C[b] ⊆ C[a]. Now, by Lemma 1.2 (2), we know

a · b ≤ a · (a → e) ≤ e.

Since b ≤ a by construction, we see that b ≤ a ≤ b → e; hence, a ∈ C[b] byconvexity.

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Let L be a member of CRL and let S ⊆ L. In what follows, we will let〈S〉 denote the submonoid of (L, ·, e) generated by S.

Lemma 2.7. Let L be a member of CRL. If S is a subset of the negativecone of L, then

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C[S] = {x ∈ L : h ≤ x ≤ h → e , ∃ h ∈ 〈S〉}.

Proof. If S ⊆ L−, then clearly 〈S〉 ⊆ L−. Thus, if we let

K = {x ∈ L : h ≤ x ≤ h → e , ∃ h ∈ 〈S〉},

then it is clear that S ⊆ K ⊆ C[S]. It will therefore suffice to show that K isa convex subalgebra of L. For all a, b ∈ K, note that there exist ha, hb ∈ 〈S〉such that

ha ≤ a ≤ ha → e and hb ≤ b ≤ hb → e.

Clearly, we may replace both ha and hb by h = ha · hb. With this in mind, itis easy to see that K is convex. Indeed, if a ≤ x ≤ b, then h ≤ a ≤ x ≤ b ≤h → e; and we see that x ∈ K.

Likewise it is easy to verify that

h ≤ a ∧ b ≤ a ∨ b ≤ h → e,

so K is closed with regard to meets and joins. Now, by Lemma 1.2 (5), wesee at once that

h2 ≤ a · b ≤ (h → e) · (h → e) ≤ h2 → e.

Hence, K is also closed under multiplication. To see that K is closed underthe arrow, first observe that a · h ≤ e and h ≤ b together imply that h2 ≤a → b. On the other hand, h ≤ a implies

a → b ≤ h → b ≤ h → (h → e) = h2 → e

by Lemma 1.2 (3).2

Corollary 2.8 Let L be a member of CRL. If a is a member of the negativecone of L, then

C[a] = {x ∈ L : an ≤ x ≤ an → e , ∃ n < ω}.

Our next goal will be to prove that every finitely generated convex sub-algebra of a commutative, residuated lattice is principal.

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Theorem 2.9. If L is a member of CRL, then SubC(L) is an algebraic, dis-tributive lattice whose compact elements are the principal convex subalgebrasof L. Moreover, the compact elements of SubC(L) form a sublattice, withjoins and meets given by the following formulas for all a, b ∈ L−:

C[a] ∩ C[b] = C[a ∨ b] and C[a] ∨ C[b] = C[a ∧ b].

Proof. We just need to establish the two equalities in the statement of thetheorem. Let a, b ∈ L−. Note that the inequalities a, b ≤ a ∨ b ≤ e implythat a ∨ b ∈ C[a] ∩ C[b] and hence C[a ∨ b] ⊆ C[a] ∩ C[b]. To obtain thereverse inclusion, suppose x ∈ C[a] ∩ C[b]. Then, by Corollary 2.8, thereexist m, n < ω such

• am ≤ x ≤ bm → e, and

• bn ≤ x ≤ bn → e.

Thus, by Lemma 2.5 (4), (a ∨ b)mn ≤ am ∨ bn ≤ x; and, likewise,

x ≤ (am → e) ∧ (bn → e) ≤ (a ∨ b)mn → e.

Hence, x ∈ C[a ∨ b]. We have shown that C[a] ∩ C[b] = C[a ∨ b].Evidently, a ∧ b ∈ C[a] ∨ C[b] = C[{a, b}] Hence, C[a ∧ b] ⊆ C[a] ∨ C[b].

To obtain the reverse inclusion, observe that, by Lemma 2.5 (1), we know

a ≤ a → e ≤ (a ∧ b) → e.

Thus, since a∧ b ≤ a, we see that a ∈ C[a∧ b]. In like manner, we can provethat b ∈ C[a ∧ b]. It follows that C[a] ∨ C[b] ⊆ C[a ∧ b].

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3 The Subvariety of CRL Generated by Resid-

uated Chains

As mentioned at the beginning of Section 2, let CRLc denote the subvarietyof CRL generated by residuated chains. Consider the following identities:

C1 e ≤ (a → b) ∨ (b → a)

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C2 e ∧ (a ∨ b) = (e ∧ a) ∨ (e ∧ b)

Theorem 3.1 The two identities above, together with those defining CRL,form an equational basis for CRLc.

Proof. Let V denote the subvariety of CRL determined by identities C1 andC2. Since any commutative, residuated chain clearly satisfies these identities,it follows that every member of CRLc is a member of V . To prove the converse,it will suffice to prove that every subdirectly irreducible member of V is totallyordered. We prove the contrapositive. Suppose that L satisfies identities C1

and C2 but is not totally ordered. Let a and b be incomparable elements inL. It follows that e 6≤ a → b and e 6≤ b → a.

Let u = e ∧ (a → b) and let v = e ∧ (b → a). By choice of a and b, weinfer that e 6= u and e 6= v. By identity C2, u ∨ v = e ∧ [(a → b) ∨ (b → a)];hence, by identity C1, u ∨ v = e. However, by Theorem 2.9, this impliesC[u] ∩ C[v] = {e}. It follows that Con(L) cannot have a monolith. Thus Lcannot be subdirectly irreducible.

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The preceding result generalizes Theorem 4.4 in Cornish [8]. The classCRLc properly includes the subvarieties of generalized Boolean algebras, rela-tive Stone algebras and commutative `-groups. Interesting examples of mem-bers of CRLc are all ideal lattices of Dedekind domains (see Larsen et al. [15],p. 137), which served as the initial motivation for identities C1 as well asidentities E1 and E2 below.

Identities C1 and C2 must both be satisfied to guarantee that subdirectlyirreducible members of CRL are totally ordered. To illustrate this, we considertwo simple examples.

Example 3.2 Let M3 = {⊥, a, b, e,>} denote the nondistributive diamond.The sublattice B = M3−{e} is a Boolean lattice (and therefore a Brouwerianlattice). Define a binary operation · on M3 as follows:

x · y =

{x ∧ y if x, y ∈ B

x if y = e.

This operation clearly defines an associative multiplication on M3, with mul-tiplicative identity e,which is residuated. If we let xc denote the complement

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of an element x ∈ B, then we can describe the residual for the multiplicationas follows:

x ↪→ y =

xc ∨ y if x, y ∈ B

xc if x ∈ B and y = e

y if x = e

Now, the algebra M3 = (M3, ·,∧,∨, ↪→, e) is simple and therefore subdirectlyirreducible. Furthermore, it is easy to see that M3 satisfies identity C1 butnot identity C2.

Example 3.3 On the other hand, consider the nonmodular pentagon N5 ={⊥, a, b, e,>}, where ⊥ < a < b < >. The residuated lattice N5 obtainedas in Example 1.1 is simple and therefore subdirectly irreducible. Moreover,it easily seen to satisfy identity C2 (even though its lattice reduct is notdistributive) but not identity C1.

Consider the following identities:

E1 (a ∧ b) → c = (a → c) ∨ (b → c)

E2 a → (b ∧ c) = (a → b) ∨ (a → c).

It is not difficult to verify that any member of CRL satisfying either E1

or E2 also satisfies C1. Conversely, in view of Theorem 3.1, any lattice thatsatisfies C1 and C2 also satisfies E1 and E2, since the last two identities holdin residuated chains. In Theorem 3.4 below, we provide an elementary proofof this fact that does not rely on the Axiom of Choice. It is a slight variationof the proof given by Ward and Dilworth in [22] in the special case when eis the greatest element of the lattice.

Theorem 3.4. Coupled with the identities defining CRL, both {C2, E1} and{C2, E2} form alternative equational bases for CRLc.

Proof. We shall establish the equivalence of identities E1 and E2 when C2 issatisfied without appealing to Theorem 3.1. As mentioned above, identitiesE1 and E2 each imply identity C1. We first prove that identities C1 andC2 together imply identity E1. Since the arrow is anti-isotone in its firstcomponent, it follows that

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(b ∧ c) → a ≥ (b → a) ∨ (c → a).

To obtain the reverse inequality, it will suffice to prove that

e ≤ [(b ∧ c) → a] → [(b → a) ∨ (c → a)].

Let u = (b → a) ∨ (c → a). Since the arrow is isotone in its secondcomponent, Lemma 1.2 (4) implies that

(b ∧ c) → u ≥ [(b ∧ c) → a] → (b → a)

= b → ([(b ∧ c) → a] → a)

≥ b → (b ∧ c),

where the last inequality follows from Lemma 1.2 (2). Likewise,

(b ∧ c) → u ≥ [(b ∧ c) → a] → (c → a)

= c → ([(b ∧ c) → a] → a)

≥ c → (b ∧ c).

Now, since the arrow preserves meets in its second component, it follows that

b → (b ∧ c) = (b → b) ∧ (b → c ≥ e ∧ (b → c),

by Lemma 1.2 (1). Likewise, we know c → (b ∧ c) ≥ e ∧ (c → b). Therefore,it follows that

(b ∧ c) → u ≥ [e ∧ (b → c)] ∨ [e ∧ (c → b)]

= e ∧ [(b → c) ∨ (c → b)] ≥ e.

Note that we used identity C2 for the last equality above, and we used identityC1 to obtain the last inequality.

We now prove that identities C1 and C2 together imply identity E2. Sincethe arrow operation is isotone in its second component, we know at once that

(a → b) ∨ (a → c) ≤ a → (b ∨ c).

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To establish the reverse inequality, we will prove that

[a → (b ∨ c)] → [(a → b) ∨ (a → c)] ≥ e.

Let u = (a → b)∨(a → c). Since the arrow is isotone in its second component,we have by Lemma 1.2 (4), (3), and (2) that

[a → (b ∨ c)] → u ≥ [a → (b ∨ c)] → (a → b)

= a → [{a → (b ∨ c)} → b]

= [a · (a → (b ∨ c))] → b

≥ (b ∨ c) → b.

Likewise, we have [a → (b ∨ c) → u ≥ (b ∨ c) → c. Consequently,

[a → (b ∨ c)] → u ≥ [(b ∨ c) → b] ∨ [(b ∨ c) → c]

= [(b → b) ∧ (c → b)] ∨ [(c → c) ∧ (b → c)]

≥ [e ∧ (c → c)] ∨ [e ∧ (b → c)]

= e ∧ [(c → b) ∨ (b → c)] ≥ e.

Note that we used identity C2 to obtain the last equality and used identityC1 to obtain the last inequality.

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In view of Theorem 3.1 and Theorem 3.4, we know that in the presenceof identity C2, the identities E1, E2, and C1 are equivalent and imply dis-tributivity. It is possible, however, to prove this implication directly, withoutappealing to the Axiom of Choice, as we now show. Again, the proof is aslight variation of the proof given by Ward and Dilworth in [22] in the specialcase when e is the greatest element of the lattice.

Theorem 3.5 Let L be a member of CRL satisfying identity C2. If L satisfiesthe equivalent identities E1, E2, and C1, then its lattice reduct is distributive.

Proof. It suffices to prove that, for all a, b, c ∈ L, a ≤ (b ∨ c) impliesa = (a ∧ b) ∨ (a ∧ c). We need only establish that (a ∧ b) ∨ (a ∧ c) ≤ a; theother inequality clearly holds in any lattice.

First, observe that, for all a, b, c ∈ L, identity E1 implies

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a → [(a ∧ b) ∨ (a ∧ c)] = [a → (a ∧ b)] ∨ [a → (a ∧ c)].

Since the residual preserves meets in its second argument, we know by Lemma1.2 (1) that

a → [(a ∧ b) ∨ (a ∧ c)] ≥ [e ∧ (a → b)] ∨ [e ∧ (a → c)].

Hence, Identities C2 and E2 together imply

a → [(a ∧ b) ∨ (a ∧ c)] ≥ e ∧ [a → (b ∨ c)]

.Now, if we assume that a ≤ (b ∨ c), then a → (b ∨ c) ≥ e. Consequently, ifa ≤ b ∨ c, we see that

a → [(a ∧ b) ∨ (a ∧ c)] ≥ e,

which, of course, implies that a ≤ (a ∧ b) ∨ (a ∧ c).2

We note that in the presence of identity C2, the equivalent to identitiesC1, E1, and E2 imply the identity a · (b ∧ c) = a · b ∧ a · c. The converse isnot true. For example, multiplication distributes over finite intersections inthe ideal lattice of a Prufer domain (see Larsen et al. [15], p. 137); however,not every such domain is a Dedekind domain.

4 More About the Structure of Con(L)

In Section 2, we provided an element-wise description for the congruencesof any member of L of CRL by establishing an isomorphism between Con(L)and SubC(L). In the last section, we used this correspondence to characterizein terms of identities those members of CRL which are subdirect products ofchains. We conclude this paper by proving that members of the variety CRLc

enjoy an additional property — their compact congruences form relativelynormal lattices.

A lattice L = (L,∧,∨) is relatively normal provided

1. L is a lower-bounded, distributive lattice, and

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2. the prime-ideals of L form a root-system under set-inclusion.

Recall that a poset is a root-system provided its principal uppersets arechains. Relatively normal lattices form an important class of lattices in theirown right. Dual relative Stone lattices and the lattice of open sets of anyhereditarily normal topological space are relatively normal, as are the latticesof compact congruences for many well-studied ordered algebraic structuressuch as representable `-groups, Reisz spaces, f -rings, and Stone algebras.(The reader is referred to Snodgrass and Tsinakis [18] and Hart and Tsinakis[12] for details and an extensive bibliography on this class.)

Since the introduction of this class in the 1950’s, a number of conditionson lower-bounded, distributive lattices equivalent to relative normality havebeen obtained. The next result, due essentially to Monteiro [17], cataloguesseveral of these conditions. Its proof is left to the reader. For our purposes,we shall find Condition (4) most useful.

Lemma 4.1 For a lower-bounded, distributive lattice L, the following areequivalent:

1. L is relatively normal;

2. Any two incomparable prime ideals in L have disjoint open neighbor-hoods in the Stone space of L;

3. The join of any pair of incomparable prime filters in L is all of L;

4. For all a, b ∈ L, there exist u, v ∈ L such that u ∧ v = ⊥ and u ∨ b =a ∨ b = a ∨ v.

We hasten to advise the reader that many authors who consider relativenormality express it as a property of the closed sets of a topological space(whereas we have used open sets); they will use and prove the duals of thestatements in Lemma 4.1.

Theorem 4.2 If L be a member of CRLc, then the principal convex subalge-bras of L form a relatively normal lattice.

Proof. Throughout the proof, we will use Theorem 2.9 repeatedly with-out explicit reference. The compact, convex subalgebras of L form a lower-bounded, distributive lattice. Let C(L) denote this lattice, and let X, Y ∈C(L). There exist a, b in the negative cone of L such that X = C[a] andY = C[b]. Let u and v be defined by

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• u = e ∧ (b → a)

• v = e ∧ (a → b).

We know know that C[u] ∩ C[v] = C[u ∨ v]. Consequently, since

u ∨ v = e ∧ [(a → b) ∨ (b → a)] = e,

by identities C2 and C1, we see that C[u] ∩ C[v] = {e}. To complete theproof, we need to show that

C[a] ∨ C[v] = C[a] ∨ C[b] = C[u] ∨ C[b].

We will prove that C[a] ∨ C[v] = C[a] ∨ C[b]; proof of the other equality issimilar.

To prove that C[a] ∨ C[v] ⊆ C[a] ∨ C[b], it suffices to prove that a ∧ v ∈C[a] ∨ C[b]. Since a ≤ e by assumption, we know a ∧ v = a ∧ (a → b). Bydefinition, C[a] ∨C[b] = C[{a, b}]; hence, it is clear that a ∧ v ∈ C[a] ∨C[b].

To obtain the reverse inclusion, it suffices to prove that a∧b ∈ C[a]∨C[v].Since multiplication distributes over finite meets (see comments at the endof Section 2), we have

(a ∧ v)2 = a2 ∧ [a · (a → b)] ∧ (b → a)2.

Since a2 ≤ a by assumption, and since a · (a → b) ≤ b by Lemma 1.2 (2), itfollows that (a ∧ v)2 ≤ a ∧ b. Since a ∧ b ≤ e by assumption, convexity nowimplies that a ∧ b ∈ C[a] ∨ C[v].

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References

[1] M. Anderson and T. Feil, Lattice-Ordered Groups, An Introduction, Rei-del, Dordrecht, Holland, 1988.

[2] R. Balbes and P. Dwinger, Distributive Lattices, Columbia, Universityof Missouri Press, 1974

[3] Garrett Birkhoff, Lattice Theory, volume 25 of Colloquium Publications,American Mathematical Society, 1967.

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[4] K. Blount and C. Tsinakis, Residuated Lattices, in preparation.

[5] T. S. Blyth, The general form of residuated algebraic structures, Bull.Soc. Math. France, 93 (1965), pp. 109-167.

[6] T. S. Blyth, Loipomorphisms, J. London Math. Soc. (2), 2 (1970), 635-642.

[7] T.S. Blyth and M.F. Janowitz, Residuation Theory, Pergamon Press,1972.

[8] W.H. Cornish, Subdirect Decompositions of Semilattice-ordered semi-groups,Mathematica Japonica, 18 (1973), 203-209.

[9] R.P. Dilworth, Abstract residuation over lattices, Bull. Amer. Math.Soc., 44 (1938), 262-268.

[10] R.P. Dilworth, Non-commutative residuated lattices, Trans. Amer.Math. Soc., 46 (1939), 426-444.

[11] G. Gierz, K. Hoffmann, K. Keimel, J. Lawson, M. Mislove, and D.Scott, A Compendium of Continuous Lattices, Springer-Verlag, BerlinHeidleberg New York, 1980.

[12] J.B. Hart and C. Tsinakis, Decompositions for relatively normal lattices,Trans. Amer. Math. Soc., 341 (1994), 519 - 548.

[13] A. Horn, Logic with truth values in a linearly ordered Heyting algebra,J. Symbolic Logic, 34 (1969), 395-408.

[14] P. Kohler, Varieties of Brouwerian algebras, Mitt. Math. Sem. Geissen.Selbstverlag des mathematischen seminars, 1975.

[15] M. Larsen and P. McCarthy, Multiplicative Theory of Ideals, AcademicPress New York and London, 1971.

[16] P. J. McCarthy, Homomorphisms of certain commutative lattice orderedsemigroups,Acta Sci. Math. (Szeged) 27 (1966), 63-65.

[17] A. Monteiro, L’arithmetique des filtres et les espaces topologiques, DeSegundo Symposium de Matematicas-Villavicencio (Mendoza, BuenosAires) (1954), 129-162.

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[18] J. Snodgrass and C. Tsinakis, Finite-valued algebraic lattices, AlgebraUniversalis 30 (1993), 311-318.

[19] M. Ward, Structure Residuation, Annals of Math., 39 (1938), 558-568.

[20] M. Ward, Residuation in structures over which a multiplication is de-fined, Duke Math. Journal, 3 (1937), 627-636.

[21] M. Ward and R. P. Dilworth, Residuated lattices, Proceedings of theNational Academy of Sciencs, 24 (1938), 162-164.

[22] M. Ward and R. P. Dilworth, Residuated lattices, Trans. Amer. Math.Soc., 45 (1939), 335-354.

1. James Hart, Box 34, Middle Tennessee State University, MurfreesboroTN 37132, (615)-898-2669, FAX (615) 898-5422, email: jhart@mtsu.edu

2. Lori Rafter, Department of Mathematics, Vanderbilt University, NashvilleTN 37235, (615) 322-6672

3. Constantine Tsinakis, Department of Mathematics, Vanderbilt Univer-sity, Nashville TN 37235, (615) 322-6672, FAX (615) 343-0215, email:constantine.tsinakis@vanderbilt.edu

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