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On the Structure of Idempotent Residuated Lattices 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; ¤; ¨; 1; {; zq such that pA; ¤q is a poset, pA; ¨; 1q is a monoid and {; z are the right and left residuals of ¨, i.e., the residuation property x ¨ y ¤ z ðñ x ¤ z{y ðñ y ¤ 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 ¤ xy ¤ x _ y. If 1 ¤ xy, then xy “ x _ y. If xy ¤ 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 ¤ x; y, then x ¤ y ðñ y Ď x. If x; y ¤ 1, then x ¤ 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 ¤ xy or xy ¤ 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; ¤y 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; ¤y 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; ¤y 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 ¤ x; y, then x ¤ y ðñ y Ď x, 3 for all x; y P C, if x; y ¤ 1, then x ¤ y ðñ x Ď y. Theorem The monoidal order Ď of any commutative idempotent residuated chain A is total and compatible with xA; ¤y. Jipsen, Gil-Ferez, Metcalfe, Tuyt, Valota Idempotent residuated lattices May 20, 2019 11 / 40 Theorem For any chain C “ xC; ¤y 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; ¤y satisfying y ¨ γapxq ¤ γapy ¨ xq. When A is commutative, the map γa is a nucleus on xA; ¤y. 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 ¤ 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 ¤ 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 ; ¤c 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 ¨ y :ðñ a ă b or pa “ b and x ¤a yq: Then is a total order on ¨ ¤ S b X :“ tXc : c P Su: We let ^ and _ be the meet and join operations for ¨ 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 ¤ 1 # &' x _ y if 1 ă a “ b „a _ y if x ¤ 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 ; ¤c y : c P Su be a family of (disjoint) chains such that each c P S is the greatest element of Xc .
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