A Survey of Generalized Basic Logic Al- gebras

2 Nikolaos Galatos and Peter Jipsenin the study of hoops, as introduced by Büchi and Owens [BO]. This showsthat it is a natural condition, appearing not only in logic, but also in algebra.While residuation corresponds to the left-continuity of a t-norm on the unitinterval, basic logic captures exactly the semantics of continuous t-norms onthe unit interval, as was shown later.On a personal note, we would like to mention that Petr Hájek’s bookhas been influential in developing some of the theory of residuated latticesand also connecting it with the study of logical systems. The book hadjust appeared shortly before a seminar on residuated lattices started at VanderbiltUniversity, organized by Constantine Tsinakis and attended by bothauthors of this article. The naturality of the definition of divisibility and itsencompassing nature became immediately clear and generalized BL-algebras(or GBL-algebras) were born. These are residuated lattices (not necessarilycommutative, integral, contractive or bounded) that satisfy divisibility: ifx ≤ y, there exist z,w with x = zy = yw. It turns out that the representablecommutative bounded GBL-algebras are exactly the BL-algebras. In otherwords, GBL-algebras are a generalization of BL-algebras that focuses on andretains the divisibility property. Examples include lattice-ordered groups,their negative cones, (generalized) MV-algebras and Heyting algebras.Despite their generality, GBL-algebras decompose into direct products oflattice-ordered groups and integral GBL-algebras, so it is these two subvarietiesthat are of main interest. From [Mu86] it follows that MV-algebras arecertain intervals in abelian l-groups and in [GT05] it is shown that GMValgebrasare certain convex sublattices in l-groups. Also, BL-chains (thebuilding blocks of BL-algebras) are essentially ordinal sums of parts of l-groups (MV-chains and product chains). It is interesting that recent workhas shown that algebras in many natural classes of GBL-algebras (includingcommutative, as well as k-potent) are made from parts of l-groups put (inthe form of a poset product) into a Heyting algebra grid. The poset productdecomposition does not work for all GBL-algebras, but it is an open problemto find BL-algebras that are not locally parts of l-groups, if such algebrasexist.2 BL-algebras as residuated lattices2.1 Residuated latticesA residuated lattice is a structure (L, ∧, ∨, ·, \,/,1), where (L, ∧, ∨) is alattice, (L, ·,1) is a monoid and the law of residuation holds; i.e., for alla,b,c ∈ L,a · b ≤ c ⇔ b ≤ a\c ⇔ a ≤ c/b.Sometimes the expression x → y is used for x\y, while y ← x (or x y)is used for y/x. The corresponding operations are called the residuals of

A Survey of Generalized Basic Logic Algebras 3multiplication. FL-algebras are expansions (L, ∧, ∨, ·, \,/,1,0) of residuatedlattices with an additional constant operation 0.Residuated lattices and FL-algebras are called commutative if multiplicationis commutative, integral if 1 is the greatest element, representable (orsemilinear) if they are subdirect products of chains, and divisible if theysatisfy:If x ≤ y, there exist z,w such that x = zy = yw.In commutative residuated lattices we have x\y = y/x, and we denote thecommon value by x → y.An FL o -algebra, also known as a bounded residuated lattice, is an FLalgebrain which 0 ≤ x for all x. In this case 0 is also denoted by ⊥. Moreover,it turns out that ⊤ = 0/0 = 0\0 is the top element. Integral FL o -algebrasare also known as FL w -algebras, and in the presence of commutativity asFL ew -algebras. 1 For more on residuated lattices see [GJKO].2.2 BL-algebrasIt turns out that BL-algebras, as defined by P. Hájek, are exactly the representableand divisible algebras within the class of integral commutativebounded residuated lattices (i.e., within FL ew -algebras).THEOREM 1 [BT03] [JT02] A residuated lattice is representable iff it satisfies[z\((x ∨ y)/x)z ∧ 1] ∨ [w((x ∨ y)/y)/w ∧ 1] = 1.The next result then follows easily.COROLLARY 2 An integal, commutative residuated lattice is representableiff it satisfies (x → y) ∨ (y → x) = 1 (prelinearity).P. Hájek defined BL-algebras using the prelinearity condition, which capturesrepresentability in the integral, commutative case. He also used thesimplified form of divisibility x∧y = x(x→y), which we will see is equivalentto the general form in the integral commutative case; actually integrality followsfrom this particular form of divisibility, by setting x = 1. The variety(i.e. equational class) of BL-algebras is denoted by BL.Representable Heyting algebras, known as Gödel algebras, are exactly theidempotent (x 2 = x) BL-algebras. Moreover, Chang’s MV-algebras are exactlythe involutive ((x→0)→0 = x) BL-algebras. For additional subvarietiesof FL w see Table 1 and 3, as well as Figure 1.1 The subscripts e, w are from the names of the rules exchange and weakening in prooftheory that correspond to commutativity and integrality.

4 Nikolaos Galatos and Peter Jipsendivisible x(x\(x ∧ y)) = x ∧ y = ((x ∧ y)/x)xprelinear x\y ∨ y\x = 1 = x/y ∨ y/xlinear or chain x ≤ y or y ≤ xintegral x ≤ 1bottom 0 ≤ xcommutative xy = yxidempotent xx = xTable 1. Some residuated lattice axioms2.3 AlgebraizationP. Hájek defined basic logic BL via a Hilbert-style system, whose sole inferencerule is modus ponens; see Table 2.(sf) (ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) (suffixing)(int) (ϕ · ψ) → ϕ (integrality)(com) (ϕ · ψ) → (ψ · ϕ) (commutativity)(conj) (ϕ · (ϕ → ψ)) → (ψ · (ψ → ϕ)) (conjunction)(·→) ((ϕ · ψ) → χ) → (ϕ → (ψ → χ))(→·) (ϕ → (ψ → χ)) → ((ϕ · ψ) → χ)(→pl) ((ϕ → ψ) → χ) → (((ψ → ϕ) → χ) → χ) (arrow prelinearity)(bot) 0 → ϕϕ ∧ ψ := ϕ · (ϕ → ψ)(conjunction definition)ϕ ∨ ψ := [(ϕ → ψ) → ψ] ∧ [(ψ → ϕ) → ϕ] (disjunction definition)¬ϕ := ϕ → 0 (negation definition)1 := 0 → 0 (unit definition)ϕϕ → ψψ(mp)Table 2. Hajek’s basic logic.Using this system, one defines as usual the notion of proof from assumptions.If ψ is a propositional formula in the language of BL, and Φ is aset of such formulas, then Φ ⊢ BL ψ denotes that ψ is provable in BL from(non-logical, i.e., no substitution instances are allowed) assumptions Φ.The following results states that ⊢ BL is algebraizable (in the sense of Blokand Piggozzi [BP89]) with respect to the variety BL of BL-algebras. It followsdirectly from the more general algebraization of substructural logics by

A Survey of Generalized Basic Logic Algebras 5FL wpsMTLGBL wFL ewpsBLMTLGBL ewpsMVBLHAMVGA.GA 4GA 3ΠGA 2BAOFigure 1. Some subvarieties of FL w ordered by inclusion

6 Nikolaos Galatos and Peter JipsenFL wFL ewGBL wGBL ewpsMTLMTLpsBLBLHApsMVMVMV nGAGA nΠBAFL-algebras with weakening= integral residuated lattices with bottomFL w -algebras with exchange= commutative integral residuated lattices with bottomGBL-algebras with weakening= divisible integral residuated lattices with bottomGBL w -algebras with exchange= commutative GBL w -algebraspseudo monoidal t-norm algebras= integral residuated lattices with bottom and prelinearitymonoidal t-norm algebras= psMTL with commutativitypseudo BL-algebras= psMTL with divisibilitybasic logic algebras= MTL with divisibilityHeyting algebras= residuated lattices with bottom and x ∧ y = xypseudo MV-algebras= psBL with x ∨ y = x/(y\x) = (x/y)\xMV-algebras or ̷Lukasiewicz algebras= BL-algebras that satisfy ¬¬x = x= commutative pseudo MV-algebrasMV-algebras generated by n + 1-chains= subdirect products of the n + 1-element MV-chainGödel algebras or linear Heyting algebras= BL-algebras that are idempotent= Heyting algebras with prelinearityGödel algebras generated by n + 1-chains= subdirect products of the n + 1-element Heyting chainproduct algebras= BL-algebras that satisfy ¬¬x ≤ (x → xy) → y(¬¬y)Boolean algebras= Heyting algebras that satisfy ¬¬x = x= MV-algebras that are idempotentTable 3. Some subvarieties of FL w

A Survey of Generalized Basic Logic Algebras 7residuated lattices given in [GO06].As usual, propositional formulas in BL are identified with terms of BL.Recall that for a set E ∪ {s = t} of equations, E |= BL s = t (s = t is asemantial consequence of E with respect to BL) denotes that for every BLalgebraB and for every assignment f into B (i.e., for every homomorphismfrom the formula/term algebra into B), if f(u) = f(v) for all (u = v) ∈ E,then f(s) = f(t).THEOREM 3 For every set Φ ∪ {ψ} of propositional formulas and every setE ∪ {s = t} of equations,• Φ ⊢ BL ψ iff {ϕ = 1 : ϕ ∈ Φ} |= BL ψ = 1• E |= BL s = t iff {(u → v) ∧ (v → u) : (u = v) ∈ E} ⊢ BL (s → t) ∧ (t → s)• ϕ ⊣⊢ (ϕ → 1) ∧ (1 → ϕ)• s = t =||= BL (s → t) ∧ (t → s) = 1As a corollary we obtain that the lattice of axiomatic extensions of BL isdually isomorphic to the subvariety lattice of BL.3 Fuzzy logics and triangular norms: retainingrepresentabilityTruth in a fuzzy logic comes in degrees. In a so-called standard model, truthvalues are taken to be real numbers from the interval [0,1], while logicalconnectives of arity n are interpreted as functions from [0,1] n to [0,1]. Inparticular, multiplication in a standard model is assumed to be continuousand a triangular norm (t-norm), namely a binary operation on the interval[0,1] that is associative, commutative, monotone and has 1 as its unit element.It is easy to see that any such continuous t-norm defines a BL-algebrastructure on [0,1]. In fact, BL is complete with respect to continuous t-norms, as proved in [CEGT], where it is shown algebraically that the varietyof BL-algebras is generated by all continuous t-norms.̷Lukasiewicz t-norm (L(x,y) = max{x+y−1,0}), product t-norm (Π(x,y) =xy), and Gödel t-norm (G(x,y) = min{x,y}), define three special standardmodels of BL. The corresponding logics that extend BL are denoted by ̷L,GL and Π.Chang [Ch59] proved that the standard model given by L(x,y) generatesthe variety of MV-algebras. Thus, ̷L is precisely the infinite-valued̷Lukasiewicz logic and it is axiomatized relatively to BL by ¬¬ϕ → ϕ. Also,GL is Gödel logic, namely the superintuitionistic logic defined by adding toBL the axiom ϕ → ϕ 2 , and it is the smallest superintuitionistic logic that is

8 Nikolaos Galatos and Peter Jipsenalso a fuzzy logic. Finally, Π is product logic and is defined relatively to BLby ¬¬ϕ → (((ϕ → (ϕ · ψ)) → (ψ · ¬¬ψ)); see [Ci01].It is easy to see that, due to the completeness of [0,1], a t-norm is residuatediff it is left-continuous; divisibility provides right-continuity, in thiscontext. It turns out that the variety of representable FL ew -algebras is generatedby left-continuous t-norms; see [JM02]. The corresponding logic iscalled monoidal t-norm logic or MTL. Uninorm logic, on the other hand,corresponds to representable FL eo -algebras, i.e., integrality is not assumed.The introduction of BL lead to the study of such general fuzzy logics, whererepresentability is the main defining property, while divisibility is droppedalltogether. In the rest of the survey, we focus on generalizations that retainthe divisibility property.4 Generalized BL-algebras: retaining divisibility4.1 GBL-algebrasA generalized BL-algebra (or GBL-algebra for short) is defined to be a divisibleresiduated lattice. We begin with presenting some equivalent reformulationsof divisibility:If x ≤ y, there exist z,w such that x = zy = yw.LEMMA 4 [GT05] The following are equivalent for a residuate lattice L.1. L is a GBL-algebra.2. L satisfies ((x ∧ y)/y)y = x ∧ y = y(y\(x ∧ y)).3. L satisfies the identities (x/y ∧ 1)y = x ∧ y = y(y\x ∧ 1).Proof. (1) ⇔ (2): Assume that L is a GBL-algebra and x,y ∈ L. Sincex ∧ y ≤ y, there exist z,w such that x ∧ y = zy = yw. Since zy ≤ x ∧ y, wehave z ≤ (x ∧ y)/y, so x ∧ y = zy ≤ ((x ∧ y)/y)y ≤ x ∧ y, by residuation, i.e.((x ∧ y)/y)y = x ∧ y. Likewise, wo obtain the second equation. The otherdirection is obvious.(2) ⇔ (3): By basic properties of residuation, we gety ∧ x = y(y\(y ∧ x)) = y(y\y ∧ y\x) = y(1 ∧ y\x).Likewise, we get the opposite identity.Conversely assume (3). Note that for every element a ≥ 1, we have 1 =a(a\1 ∧ 1) ≤ a(a\1) ≤ 1; so, a(a\1) = 1. For a = x\x, we have a 2 = a,by properties of residuation and 1 ≤ a. Thus, a = (1/a)a 2 = (1/a)a = 1.

10 Nikolaos Galatos and Peter JipsenConsequently, negative cones of l-groups are equationally defined. Aswe mentioned, cancellative GBL-algebras in general (witout the asumptionof integrality) include l-groups as well. However, we will see that everycancellative GBL-algebra is the direct product of an l-group and a negativecone of an l-group. More generally, we will see that a similar decompositionexists for arbitrary GBL-algebras.4.3 GMV-algebrasRecall that an MV-algebra is a commutative bounded residuated lattice thatsatisfies the identity x ∨ y = (x → y) → y. This identity implies that theresiduated lattice is also integral, divisible and representable, hence MValgebrasare examples of BL-algebras.It turns out that MV-algrebras are intervals in abelian l-groups. If G =(G, ∧, ∨, ·, \,/,1) is an abelian l-group and a ≤ 1, then Γ(G,a) = ([a,1], ∧, ∨, ◦, →,1,a)is an MV-algebra, where x ◦ y = xy ∨ a, and x → y = x\y ∧ 1. (A versionwhere a ≥ 1 also yields an MV-algebra.) If M is an MV-algebra then thereis an abelian l-group G and an element a ≤ 1 such that M ∼ = Γ(G,a); see[Ch59], [Mu86].Generalized MV-algebras, or GMV-algebras, are generalizations of MValgebrasin a similar way as GBL-algebras generalize BL-algebras, and aredefined as residuated lattices that satisfy x/((x∨y)\x) = x∨y = (x/(x∨y))\x.As with GBL-algebras, GMV-algebras have alternative characterizations.LEMMA 8 [BCGJT] A residuated lattice is a GMV algebra iff it satisfiesx ≤ y ⇒ y = x/(y\x) = (x/y)\x.THEOREM 9 [BCGJT] Every GMV-algebra is a GBL-algebra.Proof. We make use of the quasi-equational formulation from the precedinglemma. Assume x ≤ y and let z = y(y\x). Note that z ≤ x and y\z ≤ x\z,hencex\z = ((y\z)/(x\z))\(y\z)= (y\(z/(x\z)))\(y\z) since (u\v)/w = u\(v/w)= (y\x)\(y\z) since z ≤ x ⇒ x = z/(x\z)= (y(y\x)\z since u\(v\w) = vu\w= z\z.Therefore x = z/(x\z) = z/(z\z) = z, as required. The proof of x = (x/y)yis similar.LEMMA 10 [BCGJT]

A Survey of Generalized Basic Logic Algebras 11(i) Every integral GBL-algebra satisfies the identity (y/x)\(x/y) = x/yand its opposite.(ii) Every integral GMV-algebra satisfies the identity x/y ∨y/x = e and itsopposite.(iii) Every integral GMV-algebra satisfies the identities x/(y∧z) = x/y∨x/z,(x ∨ y)/z = x/z ∨ y/z and the opposite ones.(iv) Every commutative integral GMV-algebra is representable.It will be shown in Section 5 that the assumption of integrality in condition(iv) is not needed. They do not have to be intervals, but they are convexsublattices in l-groups. More details can be found in [GT05], while thestandard reference for MV-algebras is [CDM00].Ordinal sum constructions and decompositions have been used extensivelyfor ordered algebraic structures, and we recall here the definition for integralresiduated lattices. Usually the ordinal sum of two posets A 0 ,A 1 is definedas the disjoint union with all elements of A 0 less than all elements of A 1(and if A 0 has a top and A 1 has a bottom, these two elements are oftenidentified). However, for most decomposition results on integral residuatedlattices a slightly different point of view is to replace the element 1 of A 0 bythe algebra A 1 . The precise definition for an arbitrary number of summandsis as follows.Let I be a linearly ordered set, and for i ∈ I let {A i : i ∈ I} be a familyof integral residuated lattices such that for all i ≠ j, A i ∩ A j = {1} and 1 isjoin irreducible in A i . Then the ordinal sum ⊕ ⋃i∈I A i is defined on the seti∈I A i by⎧⎪⎨ x ·i y if x,y ∈ A i for some i ∈ Ix · y = x if x ∈ A i \ {1} and y ∈ A j where i < j⎪⎩y if y ∈ A i \ {1} and x ∈ A j where i < j.The partial order on ⊕ i∈I A i is the unique partial order ≤ such that 1 is thetop element, the partial order ≤ i on A i is the restriction of ≤ to A i , and ifi < j then every element of A i \ {1} precedes every element of A j . Finally,the lattice operations and the residuals are uniquely determined by ≤ andthe monoid operation.It not difficult to check that this construction again yields an integralresiduated lattice, and that it preserves divisibility and prelinearity. If I ={0,1} with 0 < 1 then the ordinal sum is simply denoted by A 0 ⊕ A 1 . Theassumption that 1 is join-irreducible can be omitted if A 1 has a least elementm, since if x ∧ 0 y = 1 in A 0 then x ∧ y still exists in A 0 ⊕ A 1 and has value

12 Nikolaos Galatos and Peter Jipsenm. If 1 is join-reducible in A 0 and if A 1 has no minimum then the ordinalsum cannot be defined as above. However an “extended” ordinal sum maybe obtained by taking the ordinal sum of (A 0 ⊕ 2) ⊕ A 1 , where 2 is the2-element MV-algebra.The following representation theorem was proved by Agliano and Montagna[AM03].THEOREM 11 Every linearly ordered commutative integral GBL-algebra Acan be represented as an ordinal sum ⊕ i∈I A i of linearly ordered commutativeintegral GMV-algebras. Moreover A is a BL-algebra iff I has a minimum i 0and A i0 is bounded.Thus ordinal sums are a fundamental construction for BL-algebras, and commutativeintegral GMV-algebras are the building blocks.4.4 Pseudo BL-algebras and pseudo MV-algebrasPseudo BL-algebras [DGI02] [DGI02b] are divisible prelinear integral boundedresiduated lattices, i.e., prelinear integral bounded GBL-algebras. Similarly,pseudo MV-algebras are integral bounded GMV-algebras (in this case prelinearityholds automatically). Hence BL-algebras and MV-algebras are exactlythe commutative pseudo BL-algebras and pseudo MV-algebras respectively,but the latter do not have to be commutative. They do not have to be representableeither, as prelinearity is equivalent to representability only under theassumption of commutativity. By a fundamental result of [Dv02], all pseudoMV-algebras are obtained from intervals of l-groups, as with MV-algebras inthe commutative case. Hence any nonrepresentable l-group provides examplesof nonrepresentable pseudo MV-algebras. The tight categorical connectionsbetween GMV-algebras and l-groups (with a strong order unit) haveproduced new results and interesting research directions in both areas [Ho05],[DH07], [DH09].The relationship between noncommutative t-norms and pseudo BL-algebrasis investigated in [FGI01]. In particular it is noted that any continuous t-normmust be commutative. Hájek [Ha03a] shows that noncommutative pseudoBL-algebras can be constructed on “non-standard” unit intervals, and in[Ha03b] it is shown that all BL-algebras embed in such pseudo BL-algebras.As for Boolean algebras, Heyting algebras and MV-algebras, the constant0 can be used to define unary negation operations −x = 0/x and ∼x = x\0.For pseudo BL-algebras, these operations need not coincide. Pseudo MValgebrasare involutive residuated lattices which means the negations satisfy∼−x = x = −∼x. However, for pseudo BL-algebras this identity need nothold, and for quite some time it was an open problem whether the weakeridentity ∼−x = −∼x might also fail. Recently it was noted in [DGK09] that

A Survey of Generalized Basic Logic Algebras 15RLRRLGBLIRLGMVRGBLIRRLIGBLCGBLRGMVIGMVIRGBLCRGBLCIGBLBH BrWHIRGMVLG LG − RBrRLG RLG − RBr 4RBr 3RBr 2CLG CLG − GBAOFigure 2. Some subvarieties of RL ordered by inclusion

16 Nikolaos Galatos and Peter Jipsen5 Decomposition of GBL-algebrasThe structure of l-groups has been studied extensively in the past decades.The structure of GMV-algebras has essentially been reduced to that of l-groups (with a nucleus operation), and GMV-algebras are parts of l-groups.The study of the structure of GBL-algebras has proven to be much moredifficult. Nevertheless, the existing results indicate that ties to l-groups exist.We first show that the study of GBL-algebras can be reduced to the integralcase, by showing that every GBL-algebra is the direct product of an l-groupand an integral GBL-algebra. This result was proved in the dual setting ofDRl-monoids in [Ko96] and independently in the setting of residuated latticesin [GT05]. With hindsight, the later result can, of course, be deduced byduality from the former. The presentation below follows [GT05].An element a in a residuated lattice L is called invertible if a(a\1) = 1 =(1/a)a, and it is called integral if 1/a = a\1 = 1. We denote the set ofinvertible elements of L by G(L) and the set of integral elements by I(L).Recall that a positive element a is an element that satisfies a ≥ 1.Note that a is invertible if and only if there exists an element a −1 suchthat aa −1 = 1 = a −1 a. In this case a −1 = 1/a = a\1. It is easy to see thatmultiplication by an invertible element is an order automorphism.LEMMA 13 [GT05] Let L be a GBL-algebra.(i) Every positive element of L is invertible.(ii) L satisfies the identities x/x = x\x = 1.(iii) L satisfies the identity 1/x = x\1.(iv) For all x,y ∈ L, if x ∨ y = 1 then xy = x ∧ y.(v) L satisfies the identity x = (x ∨ 1)(x ∧ 1).Proof. For (i), note that if a is a positive element in L, a(a\1) = 1 = (1/a)a,by definition; that is, a is invertible. For (ii), we argue as is the proof ofLemma 4.By (ii) and residuation, we have x(1/x) ≤ x/x = 1, hence 1/x ≤ x\1.Likewise, x\1 ≤ 1/x.For (iv), we have x = x/1 = x/(x∨y) = x/x∧x/y = 1∧x/y = y/y∧x/y =(y ∧ x)/y. So, xy = ((x ∧ y)/y)y = x ∧ y.Finally, by Lemma 4, (1/x ∧ 1)x = x ∧ 1. Moreover, by (i) x ∨ 1 isinvertible and (x ∨ 1) −1 = 1/(x ∨ 1) = 1/x ∧ 1. Thus, (x ∨ 1) −1 x = x ∧ 1, orx = (x ∨ 1)(x ∧ 1).

A Survey of Generalized Basic Logic Algebras 17The following theorem shows that if L is a GBL-algebra then the sets G(L)and I(L) are subuniverses of L. We denote the corresponding subalgebrasby G(L) and I(L).THEOREM 14 [GT05] Every GBL-algebra L is isomorphic to G(L) ×I(L).Proof. We begin with a series of claims.Claim 1: G(L) is a subuniverse of L.Let x,y be invertible elements. It is clear that xy is invertible. Additionally,for all x,y ∈ G(L) and z ∈ L, z ≤ x −1 y ⇔ xz ≤ y ⇔ z ≤ x\y. It followsthat x\y = x −1 y, hence x\y is invertible. Likewise, y/x = yx −1 is invertible.Moreover, x ∨ y = (xy −1 ∨ 1)y. So, x ∨ y is invertible, since every positiveelement is invertible, by Lemma 13(i), and the product of two invertibleelements is invertible. By properties of residuation, x ∧ y = 1/(x −1 ∨ y −1 ),which is invertible, since we have already shown that G(L) is closed underjoins and the division operations.Claim 2: I(L) is a subuniverse of L.Note that every integral element a is negative, since 1 = 1/a implies 1 ≤1/a and a ≤ 1. For x,y ∈ I(L), we get:1/xy = (1/y)/x = 1/x = 1, so xy ∈ I(L).1/(x ∨ y) = 1/x ∧ 1/y = 1, so x ∨ y ∈ I(L).1 ≤ 1/x ≤ 1/(x ∧ y) ≤ 1/xy = 1, so x ∧ y ∈ I(L).1 = 1/(1/y) ≤ 1/(x/y) ≤ 1/(x/1) = 1/x = 1, so x/y ∈ I(L).Claim 3: For every g ∈ (G(L)) − and every h ∈ I(L), g ∨ h = 1.Let g ∈ (G(L)) − and h ∈ I(L). We have 1/(g ∨h) = 1/g ∧1/h = 1/g ∧1 =1, since 1 ≤ 1/g. Moreover, g ≤ g ∨ h, so 1 ≤ g −1 (g ∨ h). Thus, by theGBL-algebra identities and properties of residuation1 = (1/[g −1 (g ∨ h)])[g −1 (g ∨ h)]= ([1/(g ∨ h)]/g −1 )g −1 (g ∨ h)= (1/g −1 )g −1 (g ∨ h)= gg −1 (g ∨ h)= g ∨ h.Claim 4: For every g ∈ (G(L)) − and every h ∈ I(L), gh = g ∧ h.In light of Lemma 13(iv), g −1 h = (g −1 h∨1)(g −1 h∧1). Multiplication by gyields h = (h∨g)(g −1 h∧1). Using Claim 3, we have gh = g(g −1 h∧1) = h∧g,since multiplication by an invertible element is an order automorphism.Claim 5: For every g ∈ G(L) and every h ∈ I(L), gh = hg.

A Survey of Generalized Basic Logic Algebras 196 Further results on the structure of GBL-algebrasIn this section we briefly summarize results from a series of papers [JM06][JM09] [JM10]. The collaboration that lead to these results started when thesecond author and Franco Montagna met at a wonderful ERCIM workshopon Soft Computing, organized by Petr Hájek in Brno, Czech Republic, in2003. This is yet another example how Petr Hájek’s dedication to the fieldof fuzzy logic has had impact far beyond his long list of influential researchpublications.6.1 Finite GBL-algebras are commutativeLEMMA 17 If a is an idempotent in an integral GBL-algebra A, then ax =a ∧ x for all x ∈ A. Hence every idempotent is central, i.e. commutes withevery element.Proof. Suppose aa = a. Then ax ≤ a ∧ x = a(a\x) = aa(a\x) = a(a ∧ x) ≤ax.In an l-group only the identity is an idempotent, hence it follows fromthe decomposition result mentioned above that idempotents are central in allGBL-algebras.In fact, using this lemma, it is easy to see that the set of idempotents in aGBL-algebra is a sublattice that is closed under multiplication. In [JM06] itis proved that this set is also closed under the residuals.THEOREM 18 The idempotents in a GBL-algebra A form a subalgebra,which is the largest Brouwerian subalgebra of A.By the results of the preceding section, the structure of any GBL-algebrais determined by the structure of its l-group factor and its integral GBLalgebrafactor. Since only the trivial l-group is finite, it follows that any finiteGBL-algebra is integral. A careful analysis of one-generated subalgebras ina GBL-algebra gives the following result.THEOREM 19 [JM06] Every finite GBL-algebra and every finite pseudo-BLalgebrais commutative.Since there exist noncommutative GBL-algebras, such as any noncommutativel-group, the next result is immediate.COROLLARY 20 The varieties GBL and psBL are not generated by theirfinite members, and hence do not have the finite model property.

20 Nikolaos Galatos and Peter Jipsen〈⊤, ⊤〉BBB ∂⊥ † = ⊤Figure 3.As mentioned earlier, BL-algebras are subdirect products of ordinal sumsof commutative integral GMV-chains, and a similar result holds without commutativityfor representable GBL-algebras. So it is natural to ask to whatextent GBL-algebras are determined by ordinal sums of GMV-algebras andHeyting algebras. We briefly recall a construction of a GBL-algebra thatdoes not arise from these building blocks.For a residuated lattice B with top element ⊤, let B ∂ denote the dualposet of the lattice reduct of B. Now we define B † to be the ordinal sum ofB ∂ and B ×B, i.e., every element of B ∂ is below every element of B ×B (seeFigure 3). Note that B † has ⊤ as bottom element, so to avoid confusion, wedenote this element by ⊥ † . A binary operation · on B † is defined as follows:〈a,b〉 · 〈c,d〉= 〈ac,bd〉〈a,b〉 · u = u/au · 〈a,b〉 = b\uu · v = ⊤ = ⊥ †Note that even if B is a commutative residuated lattice, · is in general noncommutative.LEMMA 21 • For any residuated lattice B with top element, the algebraB † defined above is a bounded residuated lattice.

A Survey of Generalized Basic Logic Algebras 21• If B is nontrivial, then B † is not a GMV-algebra, and if B is subdirectlyirreducible, so is B † .• B † is a GBL-algebra if and only if B is a cancellative GBL-algebra.6.2 The Blok-Ferreirim decomposition result for normalGBL-algebrasBuilding on work of Büchi and Owens [BO], Blok and Ferreirim [BF00] provedthe following result.PROPOSITION 22 Every subdirectly irreducible hoop is the ordinal sum of aproper subhoop H and a subdirectly irreducible nontrivial Wajsberg hoop W.This result was adapted to BL-algebras in [AM03]. To discuss furtherextensions to GBL-algebras we first recall some definitions about filters andcongruences in residuated lattices.A filter of a residuated lattice A is an upward closed subset F of A whichcontains 1 and is closed under the meet and the monoid operation. A filter Fis said to be normal if aF = Fa for all a ∈ A, or equivalently if a\(xa) ∈ Fand (ax)/a ∈ F whenever x ∈ F and a ∈ A. A residuated lattice is said tobe normal if every filter of it is a normal filter. A residuated lattice is saidto be n-potent if it satisfies x n+1 = x n , where x n = x · ...·x (n times). Notethat n-potent GBL-algebras are normal ([JM09]).In every residuated lattice, the lattice of normal filters is isomorphic tothe congruence lattice: to any congruence θ one associates the normal filterF θ = ↑{x : (x,1) ∈ θ}. Conversely, given a normal filter F, the set θ F of allpairs (x,y) such that x\y ∈ F and y\x ∈ F is a congruence such that theupward closure of the congruence class of 1 is F. Hence a residuated latticeis subdirectly irreducible if and only if it has a smallest nontrivial normalfilter.In [JM09] the following result is proved.THEOREM 23 (i) Every subdirectly irreducible normal integral GBL-algebrais the ordinal sum of a proper subalgebra and a non-trivial integral subdirectlyirreducible GMV-algebra.(ii) Every n-potent GBL-algebra is commutative and integral.A variety V has the finite embeddability property (FEP) iff every finitepartial subalgebra of an algebra in V partially embeds into a finite algebraof V. The FEP is stronger than the finite model property: for a finitelyaxiomatized variety V, the finite model property implies the decidability ofthe equational theory of V, while the FEP implies the decidability of the

A Survey of Generalized Basic Logic Algebras 23THEOREM 26 Every n-potent GBL-algebra embeds into the poset product ofa family of finite and n-potent MV-chains.Every normal GBL-algebra embeds into a poset product of linearly orderedintegral bounded GMV-algebras and linearly ordered l-groups.For Heyting algebras the above theorem reduces to the well-known embeddingtheorem into the complete Heyting algebra of all upward closed subsets ofsome poset.Various properties can be imposed on poset products to obtain embeddingtheorems for other subclasses of GBL-algebras. The follow result from [JM10]collects several of them.THEOREM 27 A GBL-algebra is• a BL-algebra iff it is isomorphic to a subalgebra A of a poset product⊗i∈I A i such that(a) each A i is a linearly ordered MV-algebra,(b) I = (I, ≤) is a root system, i.e. every principal filter of I is linearlyordered, and(c) the function on I which is constantly equal to 0 is in A;• an MV-algebra iff it is isomorphic to a subalgebra A of a poset product⊗i∈I A i such that conditions (a) and (c) above hold and(d) I = (I, ≤) is a poset such that ≤ is the identity on I;• representable iff it is embeddable into a poset product ⊗ i∈I A i such thateach A i is a linearly ordered GMV-algebra and (b) holds;• an abelian l-group iff it is embeddable into a poset product ⊗ i∈I A isuch that each A i is a linearly ordered abelian l-group and condition(d) above holds;• n-potent iff it is embeddable into a poset product of linearly orderedn-potent MV-algebras;• a Heyting algebra iff it is isomorphic to a subalgebra A of a poset product⊗ i∈I A i where condition (c) holds and in addition(e) every A i is the two-element MV-algebra;• a Gödel algebra iff it is isomorphic to a subalgebra A of a poset product⊗i∈I A i where (b), (c) and (e) hold;• a Boolean algebra iff it is isomorphic to a subalgebra A of a poset product⊗ i∈I A i where (c), (d) and (e) hold.

24 Nikolaos Galatos and Peter JipsenOf course this survey covers only some of the highlights of a few researchpapers concerned with GBL-algebras and related classes. Volumes have beenwritten about BL-algebras, MV-algebras, l-groups and Heyting algebras, aswell as about many other algebras that satisfy divisibility, and the readeris encouraged to explore the literature further, starting with the referencesbelow.BIBLIOGRAPHY[AFM07] P. Aglianò, I. M. A. Ferreirim and F. Montagna, Basic hoops: an algebraic studyof continuous t-norms, Studia Logica 87 no. 1, (2007), 73–98.[AM03] P. Aglianò and F. Montagna, Varieties of BL-algebras I: general properties, Journalof Pure and Applied Algebra 181, (2003), 105–129.[AF88] M. Anderson and T. Feil, “Lattice Ordered Groups, An Introduction”, D. ReidelPublishing Company, Dordrecht, Boston, Lancaster, Tokyo 1988.[BCGJT] P. Bahls, J. Cole, N. Galatos, P. Jipsen and C. Tsinakis, Cancellative ResiduatedLattices, Algebra Universalis 50 (1), (2003), 83–106.[BF00] W. J. Blok and I. M. A. Ferreirim, On the structure of hoops, Algebra Universalis43 (2000), 233–257.[BP89] W. J. Blok and D. Pigozzi, Algebraizable Logics, Mem. Amer. Math. Soc. 396(1989).[BP94] W. J. Blok and D. Pigozzi, On the structure of Varieties with Equationally DefinablePrincipal Congruences III, Algebra Universalis 32 (1994), 545-608.[BT03] K. Blount and C. Tsinakis, The structure of residuated lattices, InternationalJournal of Algebra and Computation 13(4), (2003), 437–461.[Bo82] B. Bosbach, Residuation groupoids, Result. Math. 5 (1982) 107-122.[BO] J. R. Büchi and T. M. Owens, Complemented monoids and hoops, unpublished[Ch59]manuscript.C. C. Chang, A new proof of the completeness of the ̷Lukasiewicz axioms, Trans.Amer. Math. Soc. 93 (1959) 74–80.[CDM00] R. Cignoli, I. D’Ottaviano and D. Mundici, “Algebraic foundations of manyvaluedreasoning”, Kluwer Academic Publishers, Dordrecht, Boston, London2000.[CEGT] R. Cignoli, F. Esteva, L. Godo and A. Torrens, Basic Fuzzy Logic is the logic ofcontinuous t-norms and their residua, Soft Computing 4, (2000), 106–112.[Ci01] P. Cintula, The ̷LΠ and ̷LΠ 1 propositional and predicate logics, Fuzzy logic2(Palma, 1999/Liptovský Ján, 2000), Fuzzy Sets and Systems 124 no. 3, (2001),289–302.[DGI02] A. Di Nola, G. Georgescu and A. Iorgulescu, Pseudo BL-algebras I, Multiple-Valued Logic 8 (2002), 673–714.[DGI02b] A. Di Nola, G. Georgescu and A. Iorgulescu, Pseudo BL-algebras II, Multiple-Valued Logic 8 (2002), 715–750.[DL03] A. Di Nola and A. Lettieri, Finite BL-algebras, Discrete Mathematics 269 (2003),93–112.[Dv02] A. Dvurečenskij, Pseudo MV-algebras are intervals in l-groups, Journal of AustralianMathematical Society 70 (2002), 427–445.[Dv07] A. Dvurečenskij, Aglianò-Montagna Type Decomposition of Pseudo Hoops andits Applications, Journal of Pure and Applied Algebra 211 (2007), 851–861.[DGK09] A. Dvurečenskij, R. Giuntini and T. Kowalski, On the structure of pseudo BLalgebrasand pseudo hoops in quantum logics, Foundations of Physics (2009), DOI10.1007/s10701-009-9342-5[DH07] A. Dvurečenskij and W. C. Holland, Top varieties of generalized MV-algebrasand unital lattice-ordered groups, Comm. Algebra 35 (2007), no. 11, 3370–3390.

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