Nilpotency and the Hamiltonian property for cancellative residuated lattices
aa r X i v : . [ m a t h . L O ] M a y NILPOTENCY AND THE HAMILTONIAN PROPERTY FOR CANCELLATIVERESIDUATED LATTICES
ALMUDENA COLACITO AND CONSTANTINE TSINAKISA
BSTRACT . The present article studies nilpotent and Hamiltonian cancellative residuatedlattices and their relationship with nilpotent and Hamiltonian lattice-ordered groups. Inparticular, results about lattice-ordered groups are extended to the domain of residuatedlattices. The two key ingredients that underlie the considerations of this paper are the cat-egorical equivalence between Ore residuated lattices and lattice-ordered groups endowedwith a suitable modal operator; and Malcev’s description of nilpotent groups of a givennilpotency class c in terms of a semigroup equation.
1. I
NTRODUCTION
The present article studies nilpotent and Hamiltonian cancellative residuated latticesand their relationship with nilpotent and Hamiltonian lattice-ordered groups. In particular,results about lattice-ordered groups ( ℓ -groups) are extended to the domain of residuatedlattices. The two key ingredients that underlie the considerations of this paper are thecategorical equivalence of [36], which provides a new framework for the study of variousclasses of cancellative residuated lattices by viewing these structures as ℓ -groups with asuitable modal operator; and Malcev’s description [33] (see also [38]) of nilpotent groupsof a given nilpotency class c in terms of a semigroup equation L c (to be defined in Section3). A plethora of evidence has been accumulated during the past two decades demonstratingthe fundamental importance of ℓ -groups in the study of algebras of logic . For example, anessential result [37] in the theory of MV-algebras is the categorical equivalence betweenthe category of MV-algebras and the category of unital Abelian ℓ -groups. Likewise, thenon-commutative generalization of this result in [17] establishes a categorical equivalencebetween the category of pseudo MV-algebras and the category of unital ℓ -groups. Further,the generalization of these two results in [36] shows that one can view GMV-algebras as ℓ -groups with a suitable modal operator. The categorical equivalence in [36] mentionedabove is another example in point.In a complementary direction, the articles [31, 21, 5, 22, 32, 6, 23] have shown thatlarge parts of the Conrad Program can be profitably extended to the much wider class of Mathematics Subject Classification.
Primary 06F05; Secondary 06D35, 06F15, 03B47, 08B20.
Key words and phrases.
Cancellative residuated lattice, Categorical equivalence, Hamiltonian lattice-orderedgroup, Lattice-ordered group, Nilpotent group, Nilpotent semigroup.The research reported here was funded by the Swiss National Science Foundation (SNF) grants200021 165850 and 200021 184693, and by the EU Horizon 2020 research and innovation programme underthe Marie Skłodowska-Curie grant agreement No 689176. We use the term algebras of logic to refer to residuated lattices—algebraic counterparts of propositionalsubstructural logics—and their reducts. Substructural logics are non-classical logics that are weaker than classicallogic, in the sense that they may lack one or more of the structural rules of contraction, weakening and exchangein their Genzen-style axiomatization. They encompass a large number of non-classical logics related to computerscience (linear logic), linguistics (Lambek Calculus), philosophy (relevant logics), and many-valued reasoning. e -cyclic residuated lattices, that is, those residuated lattices satisfying the equation x \ e ≈ e /x . The term Conrad Program traditionally refers to P. Conrad’s approach to the studyof ℓ -groups, which analyzes the structure of individual ℓ -groups, or classes of ℓ -groups,by means of an overriding inquiry into the lattice-theoretic properties of their lattices ofconvex ℓ -subgroups. In the 1960s, Conrad’s articles [9, 10, 11, 12, 13, 14] pioneered thisapproach and demonstrated its usefulness.The present work builds on the aforementioned research. Nilpotent ℓ -groups are the ℓ -groups whose group reducts are (necessarily torsion-free) nilpotent groups. They sharemany important properties with Abelian ℓ -groups, including representability (semilinear-ity) and the Hamiltonian property. In particular, they satisfy the congruence extensionproperty. The notion of Hamiltonian algebra arises as a generalization of the concept ofHamiltonian group [18]. Borrowing the terminology from group theory, an ℓ -group issaid to be Hamiltonian if every convex ℓ -subgroup is normal. Hamiltonian ℓ -groups werefirst introduced implicitly in [34], and later studied extensively (see, e.g., [8, 39, 24, 3]).While Hamiltonian ℓ -groups do not form a variety ([8, Proposition 1.4]), a largest varietyof Hamiltonian ℓ -groups does exist and was identified in [39]. A significant property ofHamiltonian ℓ -groups is representability—namely, each Hamiltonian ℓ -group is a subdi-rect product totally ordered groups. Representability and the Hamiltonian property wereestablished for nilpotent ℓ -groups in [30] (see also [27] and [39], respectively).We conclude the introduction by illustrating the article’s discourse. In Section 2, wedispatch some preliminaries on residuated lattices and their convex subuniverses. In Sec-tion 3, we study the quasivariety of submonoids of nilpotent ℓ -groups. In particular, The-orem 3.5 shows that submonoids of nilpotent ℓ -groups are precisely those nilpotent can-cellative monoids that have unique roots. The theorem also provides a characterizationfor the quasivariety of submonoids of nilpotent cancellative residuated lattices. Its proofmakes use of Theorem 3.3, which provides a bridge that connects nilpotent cancellativeresiduated lattices and nilpotent ℓ -groups. The focus of Section 4 is the prelinearity prop-erty, with particular interest for some of its implications and equivalent formulations. Weshow in Theorem 4.2 that residuals in a prelinear residuated lattice preserve finite joinsin the numerator, and convert finite meets to joins in the denominator. While prelinear-ity implies semilinearity in the presence of commutativity [25], this is no longer the casefor non-commutative varieties of residuated lattices. However, Theorem 4.2 shows thatany prelinear cancellative residuated lattice has a distributive lattice reduct and reveals thedeeper reason for the distributivity of the lattice reduct of any ℓ -group.Section 5 is devoted to Hamiltonian residuated lattices. Theorem 5.2 shows that anyHamiltonian prelinear e-cyclic residuated lattice is semilinear, which implies that any pre-linear cancellative residuated lattice is semilinear (Corollary 5.4). In particular, the resultthat Hamiltonian ℓ -groups are representable is extended in Corollary 5.4 to prelinear can-cellative residuated lattices. With these results at hand, we prove that there exists a largestvariety of Hamiltonian prelinear cancellative residuated lattices (Theorem 5.5), thereby ex-tending the corresponding result for ℓ -groups. The main focus of Section 6 is the class ofnilpotent residuated lattices. First, nilpotent cancellative residuated lattices are proved tobe Hamiltonian. As a consequence, nilpotent prelinear cancellative residuated lattices aresemilinear. The arguments make use of the corresponding results for ℓ -groups, by means ofthe categorical equivalence between nilpotent cancellative residuated lattices and nilpotent ℓ -groups with a conucleus (see Theorem 3.3).Given the role that semilinearity plays in the study of Hamiltonian and nilpotent pre-linear cancellative varieties, the final section of the paper discusses varieties of semilinear ILPOTENCY AND THE HAMILTONIAN PROPERTY FOR CANCELLATIVE RESIDUATED LATTICES 3 cancellative residuated lattices. We show, inter alia , that any variety V of semilinear can-cellative integral residuated lattices defined by monoid equations is generated by residu-ated chains whose monoid reducts are finitely generated free objects in the quasivariety ofmonoid subreducts corresponding to V (Theorem 7.5). The final result of the section, The-orem 7.8, provides a more concrete description of the generating algebras of the variety of c -nilpotent semilinear cancellative integral residuated lattices, for c ∈ N .2. R ESIDUATED L ATTICES : B
ASIC C ONCEPTS
In this section we briefly recall some basic facts about residuated lattices and theirstructure; we refer to [2], [28], [20], and [35] for further details.The set of positive natural numbers is N := { , , . . . } , and Z + is the set N ∪ { } .Throughout, by ‘poset’ we mean ‘partially ordered set’. If L is a signature and L ′ ⊆ L , an L ′ -algebra A is an L ′ -subreduct of an L -algebra B if A is a subalgebra of the L ′ -reductof B . For simplicity, when L ′ is the monoid (resp. group, lattice or semilattice) signature,we sometimes refer to A as a submonoid (resp. subgroup, sublattice or subsemilattice) ofthe L -algebra B .A residuated lattice is an algebra L = h L, ∧ , ∨ , · , \ , /, e i , where h L, · , e i is a monoid, h L, ∧ , ∨i is a lattice, and \ and / are binary operations such that, for all a, b, c ∈ L ,(2.1) ab ≤ c ⇐⇒ a ≤ c/b ⇐⇒ b ≤ a \ c, where ab stands for the product a · b , and ≤ is the lattice order. We write e for the monoididentity. The operations \ and / are referred to as left residual and right residual of · , re-spectively. We refer to a as the denominator of a \ b (resp. b/a ), and to b as the numerator of a \ b (resp. b/a ). Condition (2.1) is equivalent to · being order-preserving in each argumentand, for every a, b ∈ L , the sets(2.2) { c ∈ L | a · c ≤ b } and { c ∈ L | c · a ≤ b } containing greatest elements a \ b and b/a , respectively. Residuated lattices form a varietydenoted by RL [1, 2]. Throughout, we often write t ≤ s for the equation t ∧ s ≈ t .We recall here some relevant standard facts. Proposition 2.1.
The monoid operation · of any residuated lattice preserves all existingjoins in each argument. The residuals \ and / preserve all existing meets in the numerator,and convert existing joins in the denominator into meets. Consequently, residuals preserveorder in the numerator, and reverse order in the denominator. Proposition 2.2.
Every residuated lattice satisfies the equations x \ ( y/z ) ≈ ( x \ y ) /z, x/yz ≈ ( x/z ) /y, xy \ z ≈ y \ ( x \ z ) . For any residuated lattice L , the set L − = { a ∈ L | a ≤ e } of negative elements of L (including the monoid identity) is its negative cone . It is the universe of a submonoid, anda sublattice of L , and it can be made into a residuated lattice, by defining \ L − and / L − as a \ L − b := a \ b ∧ e a/ L − b := a/b ∧ e , for a, b ∈ L − . Residuated lattices satisfying x ∧ e ≈ x are called integral . The class ofintegral residuated lattices can be equivalently defined relative to RL by the equations(2.3) x \ e ≈ e ≈ e /x. ALMUDENA COLACITO AND CONSTANTINE TSINAKIS
We call a residuated lattice cancellative if its monoid reduct is a cancellative monoid.The class of cancellative residuated lattices is a variety (see [1, Lemma 2.5]) defined rela-tive to RL by the equations:(2.4) xy/y ≈ x ≈ y \ yx. Proposition 2.3.
The equations x/x ≈ e ≈ x \ x hold in any cancellative residuatedlattice. A residuated lattice is said to be e -cyclic if it satisfies the equation x \ e ≈ e /x . Proposition 2.4.
Every cancellative residuated lattice is e -cyclic.Proof. For any residuated lattice L and a ∈ L , we have a \ ( a/a ) = ( a \ a ) /a by Proposi-tion 2.2. Thus, by Proposition 2.3, if L is cancellative, a \ e = e /a for every a ∈ L . (cid:3) If L is a residuated lattice, we write C ( L ) for the set of all convex subuniverses of L ,ordered by set inclusion. Here, a convex subuniverse is an order-convex subuniverse of L .If L is e-cyclic, C ( L ) is a distributive lattice (see, e.g., [5, Theorem 3.8]).For any S ⊆ L , we write C[ S ] for the smallest convex subuniverse of L containing S .As usual, we call C[ S ] the convex subuniverse generated by S , and write C[ a ] for C[ { a } ] .We refer to C[ a ] as the principal convex subuniverse of L generated by the element a ∈ L .If L is a residuated lattice, and a ∈ L , the absolute value | a | ∈ L − is defined as a ∧ ( e /a ) ∧ e . Note that when a ≤ e, | a | = a .If S ⊆ L , we write h S i for the submonoid generated by S in L . The following resultsare established in [5] (see Lemma 3.2, Corollary 3.3, and Lemma 3.6 in [5]). Lemma 2.5.
In any e -cyclic residuated lattice L , the followings hold: (a) For any S ⊆ L , the convex subuniverse generated by S is C[ S ] = C[ | S | ] = { c ∈ L | t ≤ c ≤ t \ e , for some t ∈ h| S |i} = { c ∈ L | t ≤ | c | , for some t ∈ h| S |i} , where | S | := {| s | : s ∈ S } . (b) For any a ∈ L , the convex subuniverse generated by a is C[ a ] = C[ | a | ] = { c ∈ L | | a | n ≤ c ≤ | a | n \ e , for some n ∈ N } = { c ∈ L | | a | n ≤ | c | , for some n ∈ N } . (c) For any a, b ∈ L , C[ | a | ∨ | b | ] = C[ a ] ∩ C[ b ] and C[ | a | ∧ | b | ] = C[ a ] ∨ C[ b ] . If L is a residuated lattice, and a, b ∈ L , we define(2.5) λ b ( a ) := ( b \ ab ) ∧ e and ρ b ( a ) := ( ba/b ) ∧ e , and refer to λ b ( a ) and ρ b ( a ) respectively as the left and right conjugate of a by b . For anyresiduated lattice L , a convex subuniverse H ∈ C ( L ) is said to be normal if for any a ∈ H and any b ∈ L , λ b ( a ) ∈ H and ρ b ( a ) ∈ H . It was proved in [2, Theorem 4.12] that thelattice N C ( L ) of convex normal subuniverses of any residuated lattice L is isomorphic toits congruence lattice Con( L ) .A lattice-ordered group (briefly, ℓ -group ) is an algebra G = h G, ∧ , ∨ , · , − , e i suchthat h G, · , − , e i is a group, h G, ∧ , ∨i is a lattice, and the group operation distributes over ILPOTENCY AND THE HAMILTONIAN PROPERTY FOR CANCELLATIVE RESIDUATED LATTICES 5 the lattice operations. The class of ℓ -groups is a variety. Here, it is identified with theterm-equivalent subvariety LG of RL defined by the equations x ( x \ e ) ≈ e ≈ ( e /x ) x. The equivalence is given by x − := x \ e = e /x , x − y := x \ y , and yx − := y/x . In anyresiduated lattice L , an element a ∈ L is invertible , that is, it has a multiplicative inverse if a ( a \ e ) = e = ( e /a ) a. Hence, the class of ℓ -groups is identified with the class of those residuated lattices in whichevery element is invertible.Residuated lattices with a commutative monoid reduct are called commutative residu-ated lattices, and form a subvariety of RL . It is standard to call Abelian those ℓ -groupswhose underlying group is commutative. Here, a monoid-subvariety of V is any varietydefined relative to V ⊆ RL by monoid equations (e.g., commutative residuated latticesform a monoid-subvariety of RL ). We also refer to V a monoid-variety .For any monoid M , we say that ≤ ⊆ M × M is a partial order on M if it is a partialorder on M and, for all a, b, c, d ∈ M , whenever a ≤ b , also cad ≤ cbd ; if the order ≤ istotal, we call it a total order on M . If the total order is residuated, we say that M admits aresiduated total order , and we sometimes write h M , ≤i for the resulting residuated lattice.It is immediate that any total order on (the monoid reduct of) a group is a residuated totalorder. Finally, a residuated lattice admits a (residuated) total order if its monoid reductadmits a residuated total order that extends its lattice order.3. S UBMONOIDS OF N ILPOTENT L ATTICE -O RDERED G ROUPS
The primary focus of this section is the quasivariety of submonoids of nilpotent ℓ -groups.The main result of this section, Theorem 3.5, provides a characterization of these monoidsand, equivalently, of submonoids of nilpotent cancellative residuated lattices. In particular,a nilpotent monoid is a submonoid of a nilpotent ℓ -group if and only if it is cancellativeand has unique roots (in the sense to be defined below).A nilpotent group is one that has a finite central series. Given c ∈ N , nilpotent groupsof class c (in short, c -nilpotent groups) are those possessing a central series of length atmost c ; they form a variety defined by the equation [[[ x , x ] , . . . , x c ] , x c +1 ] ≈ e . Thus, -nilpotent groups coincide with Abelian groups, and every c -nilpotent group, c ∈ N ,is also ( c + 1) -nilpotent.Consider now the equation L c : q c ( x, y, ¯ z ) ≈ q c ( y, x, ¯ z ) , where ¯ z abbreviates a se-quence of variables z , z , . . . , and q c ( x, y, ¯ z ) is defined as follows, for c ∈ N : q ( x, y, ¯ z ) = xyq c +1 ( x, y, ¯ z ) = q c ( x, y, ¯ z ) z c q c ( y, x, ¯ z ) . The equation L c characterizes c -nilpotent groups. Proposition 3.1.
A group is c -nilpotent if and only if it satisfies the equation L c .Proof. See, e.g., [38, Corollary 1]. (cid:3)
We call a monoid nilpotent of class c (in short, c -nilpotent) if it satisfies L c , and call aresiduated lattice nilpotent of class c (briefly, c -nilpotent) if its monoid reduct is c -nilpotent.The class of c -nilpotent residuated lattices is a monoid-variety of residuated lattices, andcommutative residuated lattices coincide with -nilpotent residuated lattices. ALMUDENA COLACITO AND CONSTANTINE TSINAKIS
A monoid M is right-reversible if M a ∩ M b = ∅ , for all a, b ∈ M . A group of (left)quotients for a monoid M is a group G that has M as a submonoid, and such that every c ∈ G is of the form c = a − b for some a, b ∈ M . By a classical result due to Ore (see,e.g., [7, Section 1.10], [16]), a cancellative monoid M has a group of quotients (unique upto isomorphism) if and only if M is right-reversible.We call a right-reversible cancellative monoid Ore , and write G ( M ) for its group ofquotients. Further, we call a residuated lattice Ore if its monoid reduct is Ore.
Proposition 3.2.
A cancellative monoid has a c -nilpotent group of quotients if and only ifit satisfies the equation L c .Proof. See, e.g., [38, Theorem 1]. (cid:3)
The preceding result implies in particular that all nilpotent cancellative residuated latticesare Ore.The categorical equivalence in [36] provides a bridge between nilpotent cancellativeresiduated lattices and nilpotent ℓ -groups. Recall that a function σ : P → P on a poset P = h P, ≤i is a co-closure operator if it is order-preserving ( x ≤ y entails σ ( x ) ≤ σ ( y ) ),contracting ( σ ( x ) ≤ x ), and idempotent ( σ ( σ ( x )) = σ ( x ) ). The image of σ will bedenoted by P σ . We say that a co-closure operator σ on a poset P is a conucleus if σ ( e ) = eand σ ( x ) σ ( y ) ≤ σ ( xy ) . If L = h L, ∧ , ∨ , · , \ , /, e i is a residuated lattice and σ a conucleuson L , then the image L σ is a join-subsemilattice and a submonoid of L . It can be madeinto a residuated lattice, with operations ∧ σ , \ σ , and / σ , defined by a ∧ σ b := σ ( a ∧ b ) , a \ σ b := σ ( a \ b ) , a/ σ b := σ ( a/b ) , for any a, b ∈ L σ (see [36, Lemma 3.1]).Let LG cn be the category with objects h G , σ i consisting of an ℓ -group G augmentedwith a conucleus σ such that the underlying group of the ℓ -group G is the group of quo-tients of the monoid reduct of σ [ G ] . The morphisms of LG cn are ℓ -groups homomor-phisms that commute with the conuclei. The category ORL of Ore residuated latticesand residuated lattice homomorphisms was shown to be equivalent to LG cn [36, Theorem4.9]. The results collected here suffice to provide a restriction of this equivalence to thecategory N c C an RL of c -nilpotent cancellative residuated lattices and residuated latticehomomorphisms, and the full subcategory N c LG cn of LG cn consisting of objects whosefirst component is a c -nilpotent ℓ -group.We will not make use of the full categorical equivalence, but keep in mind the follow-ing key idea: Every nilpotent cancellative residuated lattice L (of class c ) ‘sits’ inside auniquely determined nilpotent ℓ -group G ( L ) (of class c ) as a submonoid, and as a join-subsemilattice. Further, L can be seen as the image of G ( L ) relative to a suitable conu-cleus. Theorem 3.3.
Let L be a c -nilpotent cancellative residuated lattice. If ≤ denotes thepartial order of L , then the binary relation (cid:22) ⊆ G ( L ) × G ( L ) defined, for a, b, c, d ∈ L ,by a − b (cid:22) c − d iff there exist m, n ∈ L such that mb ≤ nd and ma = nc, is the unique partial order on G ( L ) that extends ≤ that makes G ( L ) into a c -nilpotent ℓ -group. Further, the map σ L : G ( L ) → G ( L ) ; σ L ( a − b ) = a \ b, for all a, b ∈ L, is a conucleus on G ( L ) and L = G ( L ) σ L .Proof. See [36, Lemma 4.2, Lemma 4.3, and Lemma 4.4]. (cid:3)
ILPOTENCY AND THE HAMILTONIAN PROPERTY FOR CANCELLATIVE RESIDUATED LATTICES 7
The main result of this section, Theorem 3.5 below, characterizes those cancellativemonoids that embed into nilpotent ℓ -groups, and into nilpotent cancellative residuated lat-tices. Before we proceed with its proof, we recall a few pertinent properties of nilpotentgroups. In what follows, a monoid M (or a group) is said to have unique roots if, whenever a, b ∈ M , and a n = b n for some n ∈ N , then a = b . Lemma 3.4.
The following properties hold in any nilpotent group G . (a) Every nontrivial normal subgroup of G intersects the center nontrivially. (b) The set of torsion elements of G is a normal subgroup of G . (c) If G is torsion-free, it has unique roots.Proof. See, e.g., [29, Theorem 16.2.3, Theorem 16.2.7, and Theorem 16.2.8]. (cid:3)
For any variety V of residuated lattices, we write M ( V ) for the class of monoid sub-reducts of V , that is, those monoids that are submonoids of the monoid reduct of a residu-ated lattice from V . That M ( V ) is always a quasivariety is readily seen by checking that itis closed under ultraproducts, submonoids and direct products. Theorem 3.5.
For any monoid M , the following are equivalent: (1) M is a submonoid of a c -nilpotent ℓ -group. (2) M is c -nilpotent, cancellative, and has unique roots. (3) M has a group of quotients G ( M ) , that is c -nilpotent and torsion-free. (4) M is a submonoid of a totally ordered c -nilpotent group. (5) M is a submonoid of a c -nilpotent cancellative residuated lattice.Proof. For (1) ⇒ (2), assume that M is a submonoid of a c -nilpotent ℓ -group G . That M is c -nilpotent is immediate by Proposition 3.1. It remains to show that M has unique roots.To this end, suppose that a n = b n for some n ∈ N , and a, b ∈ M . Then, a n = b n in G .Now, since G is an ℓ -group, it is torsion-free, and by Lemma 3.4(c), a = b .For (2) ⇒ (3), observe that G ( M ) exists and is c -nilpotent by Proposition 3.2. Supposenow that ( a − b ) n = e, for some a = b ∈ M , and n ∈ N . Then, a − b is in the torsionsubgroup of G ( M ) , which is normal by Lemma 3.4(b). By Lemma 3.4(a), its intersectionwith the center of G ( M ) is non-trivial, and hence, there exists a central element c − d ∈ G ( M ) such that c = d ∈ M , and ( c − d ) m = e for some m ∈ N . As c − d is a centralelement of G ( M ) , c ( c − d ) = ( c − d ) c or, equivalently, dc − = c − d . Therefore, an easyinduction on m ∈ N shows that ( c − d ) m = ( c − ) m d m = e . This implies c m = d m , which contradicts the assumption that M has unique roots, since c and d are assumed to be distinct.For (3) ⇒ (4), it suffices to observe that G ( M ) admits a total order, as it is torsion-freeand nilpotent (see [4, Theorem 2.2.4]).Now, (4) ⇒ (5) is trivial, as any totally ordered c -nilpotent group is a c -nilpotent can-cellative residuated lattice.Finally, we show (5) ⇒ (1). By assumption M is a submonoid of a c -nilpotent can-cellative residuated lattice L . Let G ( L ) be the ℓ -group of quotients of L , as defined inTheorem 3.3. Since L is a submonoid of G ( L ) , the result follows. (cid:3) ALMUDENA COLACITO AND CONSTANTINE TSINAKIS
4. P
RELINEARITY AND ITS I MPLICATIONS
The remainder of the paper will be concerned with classes of prelinear residuated lat-tices. A residuated lattice is said to be prelinear if it satisfies the following equations: ( LPL ) ( x \ y ∧ e ) ∨ ( y \ x ∧ e ) ≈ e and ( RPL ) ( x/y ∧ e ) ∨ ( y/x ∧ e ) ≈ e . This section is devoted to exploring prelinearity, with particular interest for some of itsimplications and equivalent formulations. More precisely, Theorem 4.2 below shows thatresiduals in a prelinear residuated lattice preserve finite joins in the numerator, and con-vert finite meets to joins in the denominator. While prelinearity implies semilinearity inthe presence of commutativity [25], this is no longer the case in non-commutative set-tings. However, Theorem 4.2 shows that any prelinear cancellative residuated lattice hasa distributive lattice reduct, thereby providing an alternative proof that ℓ -groups have adistributive lattice reduct.We start with a preliminary lemma. Lemma 4.1.
The following conditions are equivalent for any lattice L . (1) L is distributive. (2) For all a, b ∈ L with a ≤ b , there exists a join-endomorphism f : L → L such that f ( b ) = a and f ( x ) ≤ x , for all x ∈ L .Proof. See [1, Proposition 4.1]. (cid:3)
Consider the following pairs of equations: ( LPL2 ) ( y ∧ z ) \ x ≈ ( y \ x ) ∨ ( z \ x ) and ( LPL3 ) x \ ( y ∨ z ) ≈ ( x \ y ) ∨ ( x \ z );( RPL2 ) x/ ( y ∧ z ) ≈ ( x/y ) ∨ ( x/z ) and ( RPL3 ) ( y ∨ z ) /x ≈ ( y/x ) ∨ ( z/x ) . Parts of the next result can be found in [2, Proposition 6.10], and [1, Corollary 4.2].
Theorem 4.2.
The followings hold. (a)
Any prelinear residuated lattice satisfies ( LPL2 ) and ( LPL3 ) . (b) In any residuated lattice that satisfies e ∧ ( y ∨ z ) ≈ ( e ∧ y ) ∨ ( e ∧ z ) , the equations ( LPL ) , ( LPL2 ) and ( LPL3 ) are equivalent. (c) Any prelinear cancellative residuated lattice has a distributive lattice reduct.Proof.
For (a), consider a residuated lattice L satisfying ( LPL ) . For any a, b, c ∈ L , ( b ∧ c ) \ a ≥ ( b \ a ) ∨ ( c \ a ) . To obtain the reverse inequality, and hence conclude ( LPL2 ) , it suffices to show thate ≤ [( b \ a ) ∨ ( c \ a )] / [( b ∧ c ) \ a ] . Let u = ( b \ a ) ∨ ( c \ a ) . Then, we have u/ [( b ∧ c ) \ a ] ≥ (1) ( b \ a ) / [( b ∧ c ) \ a ]= (2) b \ [ a/ [( b ∧ c ) \ a ]] ≥ (3) b \ ( b ∧ c )= (4) ( b \ c ) ∧ ( c \ c ) ≥ (5) ( b \ c ) ∧ e , ILPOTENCY AND THE HAMILTONIAN PROPERTY FOR CANCELLATIVE RESIDUATED LATTICES 9 where (1), (3), (4), and (5) follow by (2.1) – (2.2), and by Proposition 2.1, while (2) followsby Proposition 2.2. Likewise, u/ [( b ∧ c ) \ a ] ≥ ( c \ b ) ∧ e . Hence, u/ [( b ∧ c ) \ a ] ≥ [( b \ c ) ∧ e ] ∨ [( c \ b ) ∧ e ] = e , as was to be shown.For ( LPL3 ) , observe that it is always the case that ( a \ b ) ∨ ( a \ c ) ≤ a \ ( b ∨ c ) . To establish the reverse inequality, we show that [ a \ ( b ∨ c )] \ [( a \ b ) ∨ ( a \ c )] ≥ e . Let u = ( a \ b ) ∨ ( a \ c ) . We have [ a \ ( b ∨ c )] \ u ≥ (1) [ a \ ( b ∨ c )] \ ( a \ b )= (2) [ a ( a \ ( b ∨ c ))] \ b ≥ (3) ( b ∨ c ) \ b = (4) ( b \ b ) ∧ ( c \ b ) ≥ (5) ( c \ b ) ∧ ewhere (1), (3), (4), and (5) follow by (2.1) – (2.2), and by Proposition 2.1, while (2) followsby Proposition 2.2. Likewise, [ a \ ( b ∨ c )] \ u ≥ ( b \ c ) ∧ e . Consequently, [ a \ ( b ∨ c )] \ u ≥ [( c \ b ) ∧ e ] ∨ [( b \ c ) ∧ e ] = e , and thence, the conclusion.For (b), assume L satisfies ( LPL2 ) , and let a, b, c ∈ L . Then, [( a \ b ) ∧ e ] ∨ [( b \ a ) ∧ e ] = (1) [ a \ ( a ∧ b ) ∧ e ] ∨ [ b \ ( a ∧ b ) ∧ e ]= (2) [( a \ ( a ∧ b )) ∨ ( b \ ( a ∧ b ))] ∧ e = (3) [( a ∧ b ) \ ( a ∧ b )] ∧ e ≥ (4) e ∧ e = e , where (1) and (4) follow by (2.1) – (2.2), and by Proposition 2.1, the equality (2) followsby the assumption, and (3) is a consequence of ( LPL2 ) .Finally, assume L satisfies ( LPL3 ) , and let a, b, c ∈ L . Then [( a \ b ) ∧ e ] ∨ [( b \ a ) ∧ e ] = (1) [( a ∨ b ) \ b ) ∧ e ] ∨ [( a ∨ b ) \ a ) ∧ e ]= (2) [(( a ∨ b ) \ b ) ∨ (( a ∨ b ) \ a )] ∧ e = (3) [( a ∨ b ) \ ( a ∨ b )] ∧ e ≥ (4) e ∧ e = e , where (1) and (4) follow by (2.1) – (2.2), and by Proposition 2.1, the equality (2) followsby the assumption, and (3) is a consequence of ( LPL3 ) . For (c), we show a stronger result than the one stated above, as it suffices to assume that ( LPL3 ) and x \ x ≈ e hold in L to obtain the conclusion. For any a ≤ b ∈ L , define f : L → L, f ( x ) = a ( b \ x ) . Then, that f is a join-endomorphism follows from a ( b \ ( x ∨ y )) = (1) a (( b \ x ) ∨ ( b \ y ))= (2) a ( b \ x ) ∨ a ( b \ y ) , where (1) follows by ( LPL3 ) , and (2) by Proposition 2.1. Further, we have f ( b ) = a ( b \ b ) = a by assumption, and f ( x ) ≤ x since a ≤ b = ⇒ (3) b \ x ≤ a \ x = ⇒ (4) a ( b \ x ) ≤ x, where we get (3) by Proposition 2.1, and (4) by (2.1). We conclude by Lemma 4.1. (cid:3) Theorem 4.2(c) provides an alternative proof that ℓ -groups have distributive lattice reducts. Remark 4.3.
Even though Theorem 4.2 is presented here only for ( LPL ) , ( LPL2 ) , ( LPL3 ) ,the dual arguments show the analogous results for the equations ( RPL ) , ( RPL2 ) , ( RPL3 ) .More precisely, the equations ( RPL ) , ( RPL2 ) , ( RPL3 ) are equivalent under the hypothesisof Theorem 4.2(b). Further, ( RPL3 ) and x/x ≈ e entail distributivity of the lattice reduct.Following the proof of Theorem 4.2(c), it is easy to see that every prelinear integralresiduated lattice has a distributive lattice reduct, as it satisfies ( LPL3 ) and x \ x ≈ e.Finally, in the case of cancellative (resp. integral) residuated lattices, prelinearity is equiv-alent to ( LPL3 ) and ( RPL3 ) . The left-to-right direction is immediate from Theorem 4.2(a).For the converse, observe that ( LPL3 ) and cancellativity (resp. integrality) together entaildistributivity of the lattice reduct. Therefore, by Theorem 4.2(b), ( LPL ) must hold.5. P RELINEARITY AND C ANCELLATIVITY : THE H AMILTONIAN C ASE
This section is devoted to residuated lattices whose convex subuniverses are normal. Aresiduated lattice L is said to be Hamiltonian if every convex subuniverse H of L is nor-mal, and semilinear if L is the subdirect product of totally ordered residuated lattices (wesometimes use the term ‘(residuated) chain’ to denote a totally ordered residuated lattice).A variety V of residuated lattices is Hamiltonian if every member of V is Hamiltonian andsemilinear if each subdirectly irreducible member of V is totally ordered .The result that Hamiltonian ℓ -groups are representable is extended here to prelineare-cyclic residuated lattices. More precisely, Theorem 5.2 shows that ( LPL ) and ( RPL ) provide an axiomatization for semilinearity relative to any variety of Hamiltonian e-cyclicresiduated lattices. Later, this is used to show that a largest variety of Hamiltonian prelinearcancellative residuated lattices exists, thereby extending the analogous result for ℓ -groups. Proposition 5.1.
For any residuated lattice L , the following are equivalent: (1) L is semilinear. (2) L is prelinear, and it satisfies the quasiequation: (5.1) x ∨ y ≈ e = ⇒ λ u ( x ) ∨ ρ v ( y ) ≈ e . Proof.
See [5, Theorem 5.6]. (cid:3) It is standard to call representable those ℓ -groups that are semilinear. ILPOTENCY AND THE HAMILTONIAN PROPERTY FOR CANCELLATIVE RESIDUATED LATTICES 11
The laws ( LPL ) and ( RPL ) hold in all totally ordered residuated lattices and hence in allsemilinear residuated lattices. For Hamiltonian e-cyclic residuated lattices, the conversealso holds. Theorem 5.2.
Any Hamiltonian prelinear e -cyclic residuated lattice is semilinear.Proof. Let L be a Hamiltonian e-cyclic residuated lattice satisfying the prelinearity laws,and suppose a ∨ b = e, for a, b ∈ L . Then,e = C[ a ∨ b ] = C[ a ] ∩ C[ b ] by Lemma 2.5. Since L is Hamiltonian, for any c, d ∈ L , we have λ c ( a ) ∈ C[ a ] , and ρ d ( b ) ∈ C[ b ] . Therefore, again by Lemma 2.5, C[ λ c ( a ) ∨ ρ d ( b )] = C[ λ c ( a )] ∩ C[ ρ d ( b )] ⊆ C[ a ] ∩ C[ b ]= e , and hence, λ c ( a ) ∨ ρ d ( b ) = e. (cid:3) Corollary 5.3.
Any Hamiltonian prelinear cancellative residuated lattice is semilinear.Proof.
The conclusion follows by Proposition 2.4 and Theorem 5.2. (cid:3)
Theorem 5.2 implies the result in [25] that a prelinear commutative residuated lattice issemilinear.
Corollary 5.4.
Any commutative prelinear cancellative residuated lattice is semilinear.
The class of Hamiltonian ℓ -groups is not itself an equational class. The variety of weaklyAbelian ℓ -groups, introduced in [34], is the largest variety of Hamiltonian ℓ -groups [39,Corollary 2.3]. It is defined relative to LG by the equation(5.2) ( x ∧ e ) ≤ y − ( x ∧ e ) y. We extend this result to the context of prelinear cancellative residuated lattices. Note thatthe analogous result fails for e-cyclic residuated lattices (see [5, Theorem 6.3]), as there isno largest variety of Hamiltonian e-cyclic residuated lattices.
Theorem 5.5.
There exists a largest variety of Hamiltonian prelinear cancellative resid-uated lattices. More precisely, a variety V of prelinear cancellative residuated lattices isHamiltonian if and only if V satisfies the equations (5.3) ( x ∧ e ) ≤ λ y ( x ) and ( x ∧ e ) ≤ ρ y ( x ) , where λ y and ρ y are defined as in (2.5).Proof. Suppose first that V is a variety of prelinear cancellative residuated lattices thatsatisfies equation (5.3). Let L ∈ V , H a convex subuniverse H ∈ C ( L ) , a ∈ H , and b ∈ L .We have ( a ∧ e ) ∈ H and ( a ∧ e ) ≤ ( b \ ab ) ∧ e ≤ e , ( a ∧ e ) ≤ ( ba/b ) ∧ e ≤ e . Hence the convexity of H implies that λ b ( a ) , ρ b ( a ) ∈ H . We have shown that H is normaland hence V is a Hamiltonian variety.To prove the converse direction, we use logical contrapositive. Suppose that V is avariety of prelinear cancellative residuated lattices that fails either of the equations in (5.3), say the first one. Then, by Corollary 5.4, there exists a residuated chain T ∈ V and anelement a ∈ T − such that a b \ ab ∧ e for some b ∈ T or, by cancellativity,(5.4) ab < ba for some b ∈ T. Condition (5.4) can be used to construct a non-Hamiltonian member L ∈ V . Note first thefollowing: Claim.
For any n ∈ N , a n b < ba n .Proof of Claim. We proceed by induction on n ∈ N . The base case follows from (5.4).For the induction step, observe that a n +1 b = aa n b< (1) aba n < (2) ba a n = ba n +1) , where (1) follows by the induction hypothesis, and (2) from (5.4). (cid:3) Claim.