On the structure of idempotent residuated lattices

Peter Jipsen joint work with José Gil-Ferez, George Metcalfe (U. Bern) Constantine Tsinakis (Vanderbilt U.) Olim Tuyt (U. Bern) and Diego Valota (U. Milan)

Chapman University

BLAST, U. Colorado at Boulder May 20 - 24, 2019

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 1 / 40 Outline

Monoidal preorder of idempotent residuated lattices Idempotent residuated chains Conservative residuated lattices and catalan algebras Amalgamation property for semilinear idempotent RLs Some results about idempotent involutive residuated lattices

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 2 / 40 Introduction

A residuated poset is a partially ordered algebra pA, ď, ¨, 1, {, zq such that

pA, ďq is a poset, pA, ¨, 1q is a monoid and {, z are the right and left residuals of ¨, i.e., the residuation property

x ¨ y ď z ðñ x ď z{y ðñ y ď xzz

holds for all x, y, z P A.

As usual, we abbreviate x ¨ y by xy and adopt the convention that ¨ binds stronger than {, z. If the poset is a lattice pA, ^, _q then A “ pA, ^, _, ¨, 1, {, zq is a residuated lattice.

A is idempotent if xx “ x for all x P A. Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 3 / 40 The cone of a residuated lattice

The cone of a residuated lattice A is the union of its positive cone Ò1 and negative cone Ó1. A is conical if A “Ò1 YÓ1. Lemma (Stanovsky 2007)

For any idempotent residuated lattice A and x, y P A: x ^ y ď xy ď x _ y. If 1 ď xy, then xy “ x _ y. If xy ď 1, then xy “ x ^ y. If x, y PÒ1, then xy “ x _ y, and, if x, y PÓ1, then xy “ x ^ y. xÓ1, ^, _, ñ, 1y is a Brouwerian algebra, where x ñ y :“ pxzyq ^ 1.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 4 / 40 The monoidal preorder

Define the monoidal preorder on an idempotent residuated lattice A by

x Ď y ðñ xy “ x.

Lemma

For any idempotent residuated lattice A, the relation Ď is a preorder on A with top element 1, where if A has a least element, this is also the least element of Ď. Moreover, for any x, y P A: If 1 ď x, y, then x ď y ðñ y Ď x. If x, y ď 1, then x ď y ðñ x Ď y.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 5 / 40 Conservative residuated lattices

A residuated lattice A is conservative if xy P tx, yu for all x, y P A. In semigroup theory conservative is also called quasitrivial Every conservative residuated lattice is idempotent. For a conservative residuated lattice the monoidal preorder determines xy via # x if x Ď y xy “ y otherwise.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 6 / 40 Lemma Every idempotent residuated chain A is conservative.

Proof. Since A is a chain, 1 ď xy or xy ď 1, hence xy “ x _ y or xy “ x ^ y. In a total order it follows that xy P tx, yu.

The converse only holds for the elements in the cone. Lemma

If A is a conservative residuated lattice, then xÓ1 YÒ1, ďy is a chain.

Proof. xy “ x ^ y for any x, y in the negative cone of A, and therefore conservativity implies that x ^ y “ x or x ^ y “ y, so xÓ1, ďy is a chain. The argument for the positive cone is symmetrical.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 7 / 40 Odd Sugihara monoids

A residuated lattice is commutative if xy “ yx. In this case xzy “ y{x and is written as x Ñ y. Example

The variety OSM of odd Sugihara monoids consists of all semilinear commutative idempotent residuated lattices satisfying px Ñ 1q Ñ 1 « x, and is generated as a quasivariety [M. Dunn 1970] by the algebra

Z “ xZ, ^, _, ¨, Ñ, 0y,

where 1 “ 0 and ¨ is the meet operation of the total order

... ă ´3 ă 3 ă ´2 ă 2 ă ´1 ă 1 ă 0,

A variety is called

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 8 / 40 Some subvarieties of idempotent residuated lattices

IdRL = the variety of idempotent residuated lattices CsRL = varieties generated by conservative residuated lattices SemIdRL = varieties generated by idempotent residuated chains OSM = variety of odd Sugihara monoids IdRL

CsRL CIdRL

SemIdRL CCsRL

CSemIdRL C = commutative members

OSM

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 9 / 40 Properties of idempotent residuated lattices

LF: locally finite FEP: finite embeddability property AP: amalgamation property ?: is still open

Variety LF FEP AP IdRL no ? ? CIdRL no yesvanAlten’05 yes CsRL nonew Thm(1) yesnew Cor(2) ? CCsRL ? yesnew Cor(2) ? SemIdRL nonew Thm(1) yesnew Cor(2) ? SemCIdRL yesRaftery’07 yesRaftery’07 yesnew Thm(3) OSM yesDunn’70 yesDunn’70 yesMarchioniMetcalfe’12

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 10 / 40 Let C “ xC, ďy be any chain. We say that a total order Ď on C with top element 1 is compatible with C if

1 whenever C has a least element K, also xC, Ďy has least element K, 2 for all x, y P C, if 1 ď x, y, then x ď y ðñ y Ď x, 3 for all x, y P C, if x, y ď 1, then x ď y ðñ x Ď y.

Theorem

The monoidal order Ď of any commutative idempotent residuated chain A is total and compatible with xA, ďy.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 11 / 40 Theorem

For any chain C “ xC, ďy and compatible total order Ď on C, the algebra xC, ^, _, ¨, 1y is a commutative idempotent totally ordered monoid, where # x if x Ď y, x ¨ y “ y otherwise.

Moreover, if C is finite, then ¨ has a (uniquely determined) residual Ñ and xC, ^, _, ¨, Ñ, 1y is a commutative idempotent residuated chain.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 12 / 40 [Raftery 2007] describes the structure of all finite commutative idempotent residuated chains by dividing the negative cone of such an algebra into a family of (possibly empty) intervals indexed by the positive elements. We give a more symmetric version of this result covering all chains, where both negative and positive cones are divided into families of nonempty intervals with greatest elements that together form a retract of the algebra.

For any residuated lattice A and a P A, the map γa : A Ñ A mapping x to pa{xqza is a closure operator on xA, ďy satisfying y ¨ γapxq ď γapy ¨ xq.

When A is commutative, the map γa is a nucleus on xA, ďy.

The algebra Aγa “ xAγa , ^, _γa , ¨γa , Ñ, γap1qy with Aγa “ tγapbq : b P Au,

b _γa c “ γapb _ cq, and b ¨γa c “ γapbcq is always a commutative residuated lattice.

For convenience, we define „x “ x Ñ 1.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 13 / 40 Lemma

If A is a commutative idempotent residuated chain, then Aγ1 is a retract of A. Moreover, any homomorphism f : A Ñ B between commutative idempotent residuated chains restricts to a homomorphism

f æAγ1 : Aγ1 Ñ Bγ1 .

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 14 / 40 Proposition

For any commutative idempotent residuated chain A:

1 Aγ1 is a totally ordered odd Sugihara monoid. 2 For each c P Aγ1 , the set Ac “ tx P A : γ1pxq “ cu is an interval of A with greatest element c. 3 For all x, y P A, 1 If x, y P Ac for some c P Aγ1 with c ď 1, then xy “ x ^ y. 2 If x, y P Ac for some c P Aγ1 with 1 ă c, then xy “ x _ y. 3 If x P Ac , y P Ad for some c ‰ d P Aγ1 , then xy “ x ðñ cd “ c.

4 For all x, y P A with x P Ac for some c P Aγ1 , # „c _ y if x ď y, x Ñ y “ „c ^ y if y ă x.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 15 / 40 Let S be any totally ordered odd Sugihara monoid and let X “ txXc , ďc y : c P Su be a family of (disjoint) chains such that each c P S is the greatest element of Xc . We define for all a, b P S with x P Xa and y P Xb,

x ĺ y :ðñ a ă b or pa “ b and x ďa yq. Then is a total order on ĺ ď S b X :“ tXc : c P Su. We let ^ and _ be the meet and join operations for ĺ and define the algebra S b X :“ xS b X , ^, _, ¨, Ñ, 1y, where for a, b S and x X , y X , $ P P a P b ’ ’x ^ y if a “ b ď 1 # &’ x _ y if 1 ă a “ b „a _ y if x ď y, x¨y “ ’ and x Ñ y “ ’x if a ‰ b and ab “ a „a ^ y if y ă x. %’ y if a ‰ b and ab “ b

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 16 / 40 Theorem Let S be any totally ordered odd Sugihara monoid and let X “ txXc , ďc y : c P Su be a family of (disjoint) chains such that each c P S is the greatest element of Xc . Then S b X is a commutative idempotent residuated chain satisfying X X S “ pS b qγ1 and pS b qc “ Xc for each c P S. Moreover, every commutative idempotent residuated chain has this form.

Corollary (Raftery 2007)

The variety of semilinear commutative idempotent residuated lattices is locally finite.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 17 / 40 Idempotent Residuated Chains

Since the monoidal preorder of such an idempotent residuated chain A is not a partial order in general, we define for x, y P A, x ĎĚ y ðñ x Ď y and y Ď x and x ∥ y ðñ x Ę y and y Ę x.

x P A is central if it commutes with every other element of A, i.e., xy “ yx for all y P A. Lemma For any idempotent residuated chain A, if two elements do not commute, then one is positive and the other negative. Moreover, for each x P A, there are two distinct possibilities: 1 x is central and for all y P A, either x Ď y or y Ď x, and x ĎĚ y ðñ x “ y. 2 x is not central, and there is a unique y P A such that x and y do not commute.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 18 / 40 Idempotent Residuated Chains

For any element a of an idempotent residuated chain A, define a7 “ a, if a is central, and otherwise, the only element of A that does not commute with a.

Notice that in both cases pa7q7 “ a. If a is not central, we call ta, a7u a noncommuting pair. Lemma

For any idempotent residuated chain A, a P A, and x P Azta, a7u,

a Ď x ðñ a7 Ď x and x Ď a ðñ x Ď a7.

We now identify properties of the monoidal preorder, analogously to the commutative case, and show that in the finite setting these properties provide a complete description of the algebra.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 19 / 40 Laced preorders

A preorder Ď on a set A is laced if

1 it has a (unique) top element 1, 2 each a P A is either comparable with all the other elements and we fix a7 “ a, or there is a unique element a7 such that a ĎĚ a7 or a ∥ a7, 3 for all a P A and x P Azta, a7u,

a Ď x ðñ a7 Ď x and x Ď a ðñ x Ď a7.

Now let C “ xC, ďy be any chain. We say that a laced preorder Ď on C is compatible with C if

1 any least element of C is also the least element of Ď, 2 for all x, y P C, if 1 ď x, y, then x ď y ðñ y Ď x, 3 for all x, y P C, if x, y ď 1, then x ď y ðñ x Ď y, 4 for each x P C, if x ‰ x 7, then 1 ď x ðñ x 7 ď 1.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 20 / 40 Theorem

1 The monoidal preorder Ď of any idempotent residuated chain A is laced and compatible with xA, ďy. 2 For any chain C “ xC, ^, _y and compatible laced preorder Ď on C, the algebra xC, ^, _, ¨, 1y is an idempotent totally ordered monoid, where # x if x Ď y, x ¨ y “ y otherwise, Moreover, if C is finite, then ¨ has (uniquely determined) residuals z and { and xC, ^, _, ¨, z, {, 1y is an idempotent residuated chain.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 21 / 40 We use this representation theorem to count the number of idempotent residuated chains of size n ě 2 up to isomorphism. Theorem

The number Ipnq of idempotent residuated chains of size n ě 2 is satisfies the recurrence formula Ip2q “ 1, Ip3q “ 2, Ipn ` 2q “ 2Ipnq ` 2Ipn ` 1q, hence the number of idempotent residuated chains of size n ě 2 is ` ˘ ` ˘ ? n ? n 1 ` 3 ´ 1 ´ 3 Ipnq “ ? . 2 3

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 22 / 40 Recall that the variety of semilinear commutative idempotent residuated lattices is locally finite. This property fails, if semilinearity is weakened to distributivity or idempotence is weakened to being square-increasing, square-decreasing, or n-potent for n ě 3 [Raftery 2007]. Theorem (1)

The variety of semilinear idempotent residuated lattices is not locally finite.

Proof. It suffices to exhibit an infinite idempotent residuated chain with a finite set of generators. Consider the set Z of integers with the standard order, and define x ¨ y “ x if |x| ě |y| and x ¨ y “ y otherwise. It is easy to see that this determines the unique structure of an idempotent residuated chain. Moreover, xzx “ |x| for each x P Z, and if x ą 0, then xz0 “ ´x ´ 1. So we have 1 “ | ´ 1|, ´2 “ 1z0, 2 “ | ´ 2|, ´3 “ 2z0, etc. Also, 0 “ p´1q{p´1q. Hence t´1u generates the whole algebra.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 23 / 40 Conservative Residuated Lattices

Recall that idempotent residuated lattices are conservative if p@xqp@yqpxy « x or xy « yq. A variety V has the finite embeddability property if any finite partial subalgebra of a member of V embeds into a finite member of V. Theorem Let K be a class of conservative residuated lattices defined relative to IdRL by positive universal formulas in the language t_, ¨, 1u. Then the variety V generated by K has the finite embeddability property.

Corollary (2)

CsRL, CCsRL, and SemIdRL have the finite embeddability property.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 24 / 40 Catalan sums

Any finite commutative conservative residuated lattice is subdirectly irreducible, since its negative cone is a finite chain of (central) idempotents.

The class SipCCsRLqfin of finite subdirectly irreducible members of CCsRL therefore consists of all finite commutative conservative residuated lattices and generates CCsRL. Moreover, using B. McCune’s Mace4, it can be shown that there are 1, 2, 5, 14, 42, 132, 429, 1430, 4862, and 16796 such algebras of size 2 to 11, respectively. ` ˘ 1 2n This sequence corresponds exactly to the Catalan numbers Cn “ n`1 n .

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 25 / 40 Catalan sums

For A, B P SipCCsRLqfin, define the Catalan sum C “ A ⃝c B as follows: Let C be the disjoint union of A and B and define a lattice order ďC “ ďA Y ďB Y ptKAu ˆ Bq Y pA ˆ Ò1Bq. To specify ¨C, it suffices to define the following monoidal order: ĎC “ ĎA Y ĎB Y ptKAu ˆ Bq Y pB ˆ pAztKAuqq. Informally, the monoidal order of C is the ordinal sum tKAu ‘ xB, Ďy ‘ xAztKAu, Ďy. Since the top element is always the identity, it follows that if A is nontrivial, then 1C “ 1A and otherwise 1C “ 1B. The lattice order implies that KA is the least element of C, and that a _ b “ 1B _ b whenever a P AztKAu and b P B. If A or B is a one element algebra, then the underlying lattice of C is simply the ordinal sum of the lattices of A and B. Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 26 / 40 Catalan sums

1C “ 1A B 1 AztKAu B 1C “ 1A 1B B B A K “K B KB KC “KA KA

Figure: The Catalan sum C “ A ⃝c B, for nontrivial A.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 27 / 40 Catalan algebras

Lemma

If A, B P SipCCsRLqfin, then A ⃝c B P SipCCsRLqfin.

Lemma

Suppose that C P SipCCsRLqfin has size n ě 2. Then C “ A ⃝c B for a pair A, B P SipCCsRLqfin that is unique up to isomorphism.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 28 / 40 Catalan algebras

An algebra is Catalan if it is a one-element residuated lattice or a Catalan sum of Catalan algebras. 1 2 1 ⃝ 1 In particular, if C1 is a one-element residuated lattice, then C1 “ C1 c C1 is the two element Boolean algebra. 3 1 ⃝ 2 3 2 ⃝ 1 The two three element chains are C1 “ C1 c C1 and C2 “ C1 c C1. In general, the algebras of size n are built by constructing all Catalan sums of algebras A and B of size n ´ k and k respectively, as k ranges from 1 to n ´ 1. Theorem The class of finite conservative commutative residuated lattices is precisely the class of Catalan algebras.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 29 / 40 Catalan algebras

4 3 3 2 1 1 3 4 2 1 2 3 2 3 2 2 2 1 1 2 1 1 3 1 2 3 1 1 0 0 0 0 0 0 0 0 0 0 0 1 2 3 3 4 4 4 4 4 5 5 C1 C1 C1 C2 C1 C2 C3 C4 C5 C1 C2

2 2 1 1 1 1 3 4 3 3 2 2 2 4 3 1 2 2 4 2 3 4 2 3 4 3 3 4 3 4 3 1 1 1 4 1 4 1 3 1 4 1 2 2 4 2 3 4 0 0 0 0 0 0 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 5 5 5 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14

Figure: Subdirectly irred. commutative conservative residuated lattices of size ď 5

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 30 / 40 This yields the following result. Theorem

The number of conservative` ˘ commutative residuated lattices of n ě 1 1 2pn´1q elements is Cpnq “ n n´1 , that is, the pn ´ 1qth Catalan number.

Proof. We will prove the result by induction. The sequence pCři : i ě 0q of n Catalan numbers is determined by C0 “ 1 and Cn`1 “ i“1 Ci Cn´i . Obviously, Cp1q “ 1 “ C0. Suppose now that n ą 1. Then

ÿn Cpn ` 1q “ Cpkq ¨ Cpn ` 1 ´ kq k“1 ÿn nÿ´1 “ Ck´1Cn´k “ Ci Cn´1´i “ Cn´1. k“1 i“0

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 31 / 40 The Amalgamation Property

A span of a class of algebras K is a pair of embeddings xi1 : A ãÑ B, i2 : A ãÑ Cy between algebras A, B, C P K. K has the amalgamation property if for every span of K, there exist D P K and embeddings j1 : B ãÑ D and j2 : C ãÑ D such that j1 ˝ i1 “ j2 ˝ i2. The main tools for proving AP will be the characterization of commutative idempotent residuated chains and the following criterion for amalgamation in varieties of semilinear residuated lattices. Theorem (Metcalfe, Montagna, Tsinakis 2014) Let V be a variety of semilinear residuated lattices with the congruence extension property, and let T be the class of finitely generated totally ordered members of V. If every span in T has an amalgam in V, then V has the amalgamation property.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 32 / 40 AP for commutative semilinear idempotent RLs

From the structural description of commutative idempotent residuated chains: Lemma The class of commutative idempotent residuated chains has the amalgamation property.

Since every variety of commutative residuated lattices has the congruence extension property, we get the following result. Theorem (3)

The variety of semilinear commutative idempotent residuated lattices has the amalgamation property.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 33 / 40 Does there exists a noncommutative variety of residuated lattices that has the amalgamation property? There are two nonisomorphic noncommutative idempotent residuated chains of size 4: C4 shown below, and its “opposite” (i.e., the algebra resulting from swapping the order of the product).

c7 1 1 c c ĎĚ c7 K K

Figure: The algebra C4.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 34 / 40 To show that VpC4q has the congruence extension property, we make use of some results of [van Alten 2005].

Recall that a term up⃗x,⃗y q is an ideal term in ⃗x for a variety V with respect to a (term-definable) constant 1 if and only if V |ù tp⃗1,⃗y q « 1. The ideals (with respect to 1) of an algebra A P V are the subsets I Ď A such that up⃗a,⃗b q P I for every ideal term up⃗x,⃗y q and ⃗a P I, ⃗b P A. If A is a residuated lattice, then the ideals with respect to 1 coincide with the convex normal subalgebras of A. [van Alten 2005] proved that the variety generated by a residuated lattice A has equationally definable principal congruences (and therefore the congruence extension property) if there exists a finite set J of ideal terms (with respect to 1) such that for all a, b PÓ1, there exists upx, yq P J satisfying b P xayA ðñ b “ uApa, bq, where xayA denotes the convex normal subalgebra generated by a.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 35 / 40 Lemma

VpC4q has the congruence extension property.

Proof.

Observe first that C4 has only the trivial proper subalgebra, since c7 “ cz1 “ Kz1, K“ c7z1 “ 1{c7, and c “ 1{c7. It suffices now to check that C4 satisfies (35) for the set of ideal terms J “ t1, x, pxz1qz1, 1{p1{xqu. For a “ 1, we have x1yC4 “ t1u “ Jp1q, and C4 for a ‰ 1, we have xay “ C4 and Jpaq “ t1, c, Ku.

Theorem

VpC4q has the amalgamation property.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 36 / 40 Involutive residuated lattices

An involutive residuated lattice xA, ^, _, ¨, 1, {, z, „, ´y is a residuated lattice with two linear negations that satisfy „x “ xz0, ´x “ 0{x, „´x “ x “ ´„x where 0 “ ´1 “„1 An involutive residuated lattice is cyclic if „x “ ´x. ¨ commutative ùñ cyclicity. An integral (i.e. 1 “ top element) idempotent involutive residuated lattice is a Boolean algebra. Sugihara chains are the only commutative idempotent involutive residuated chains. Theorem Every cyclic idempotent involutive residuated lattice is commutative. Every finite idempotent involutive residuated chain is commutative.

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 37 / 40 A noncyclic idempotent involutive residuated lattice

Let A “ Z ‘ t1u ‘ ZB, where ‘ is the ordinal sum. Lattice order:

¨ ¨ ¨ a´2 ă a´1 ă a0 ă a1 ă a2 ¨ ¨ ¨ ă 1 ă ¨ ¨ ¨ b2 ă b1 ă b0 ă b´1 ă b´2 ¨ ¨ ¨ Compatible monoid preorder:

¨ ¨ ¨ a´2 ĎĚ b´2 Ă a´1 ĎĚ b´1 Ă a0 ĎĚ b0 Ă a1 ĎĚ b1 Ă a2 ĎĚ b2 Ă ¨ ¨ ¨ Ă 1 Linear negations:

1 “ 0, „ai “ bi , „bi “ ai´1, ´ai “ bi`1, ´bi “ ai

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 38 / 40 A typical finite commutative idempotent involutive RL

1 “ 0 8 2 4 3 9 2 4 5 1 6 5 6 3 10 7 8 7 9 10 K K

ďA ĎA

Figure: The two partial orders ďA and ĎA of an algebra A P IdInRL Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 39 / 40 Commutative IdInRLs are disjoint unions of Boolean algebras

In an involutive residuated lattice, idempotence implies that 0 ď 1 and that pr0, 1s, ¨, `, ´, 0, 1q is a Boolean algebra, where x ` y “ „p´y ¨ ´xq.

For A in CIdInRP, define the terms 0x “ ´x ¨ x and 1x “ ´p´x ¨ xq, and let rra, bss “ tc P A : ac “ a, bc “ cu. Theorem

The semilattice intervals prr0x , 1x ss, ¨, `, ´, 0x , 1x q are also Boolean algebras and they partition A.

We conjecture that all finite idempotent involutive residuated lattices are commutative and that the monoidal semilattice orders are distributive (checked up to size 16).

Thanks!

Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 40 / 40