Shanks Workshop Ordered Algebras and Logic February 22-23, 2014 – Booklet of Abstracts – National Science Foundation, Shanks Endowment, Vanderbilt University, Consortium for Order Algebra and Logic Venue Department of Mathematics of Vanderbilt University 1326 Stevenson Center Ground floor, Room 1308 Organizers José Gil-Férez, University of Cagliari, Italy Constantine Tsinakis, Vanderbilt University, USA Hongjun Zhou, Shaanxi Normal University, China ii Schedule Saturday, February 22 09:30 - 10:15 10:30 - 11:15 Francesco Paoli Towards Multiset Consequence Relations Michal Botur Tense MV-algebras 11:15 - 12:45 Lunch 12:45 - 13:30 Antonio Ledda The Failure of Amalgamation Property for Semilinear Varieties of Residuated Lattices Hongjun Zhou Stone Duality for R0 -algebras (Nilpotent Minimum Algebras) with Internal States William Young Heyting Algebras as Intervals of Commutative, Cancellative Residuated Lattices 13:45 - 14:30 14:45 - 15:30 17:30 - 20:00 Reception Sunday, February 23 09:30 - 10:15 10:30 - 11:15 Tomasz Kowalski Kites James B Hart Completions of Partial Information Systems 11:15 - 12:45 Lunch 12:45 - 13:30 Jan Kühr On Varieties of Lattice-ordered Effect Algebras Luca Spada Canonical Formulas and Residuated Lattices José Gil-Férez Skid Decomposition of Algebras: Generalizing Direct Decomposition Franco Montagna A Categorical Equivalence for Product Algebras 13:45 - 14:30 14:45 - 15:30 15:45 - 16:30 iii Index of Abstracts Tense MV-algebras Michal Botur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Skid Decomposition of Algebras: Generalizing Direct Decomposition José Gil-Férez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Completions of Partial Information Systems James B Hart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Kites Tomasz Kowalski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 On Varieties of Lattice-ordered Effect Algebras Jan Kühr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Failure of Amalgamation Property for Semilinear Varieties of Residuated Lattices Antonio Ledda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A Categorical Equivalence for Product Algebras Franco Montagna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Towards Multiset Consequence Relations Francesco Paoli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Canonical Formulas and Residuated Lattices Luca Spada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Heyting Algebras as Intervals of Commutative, Cancellative Residuated Lattices William Young . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 iv Stone Duality for R0 -algebras (Nilpotent Minimum Algebras) with Internal States Hongjun Zhou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 v Shanks Workshop Ordered Algebras and Logic Tense MV-algebras Michal Botur Palachý University in Olomouc, Czech Republic [email protected] The main aim of the talk is to present tense MV-algebras which are just MV-algebras with new unary operations G and H which express a universal time quantifiers. Tense MV-algebras were introduced by D. Diaconescu and G. Georgescu. Using a new notion of an fm-function between MV-algebras we show that any tense semisimple MV-algebra is induced by a frame analogously to classical works in this field of logic. As a by-product we obtain a new characterization of extremal states on MV-algebras. Our method gives a general framework for representing functions (including MV-morphisms) between MV-algebras. Skid Decomposition of Algebras: Generalizing Direct Decomposition José Gil-Férez University of Cagliari, Italy [email protected] (Joint work with Antonio Ledda, Francesco Paoli, and Antonino Salibra) We introduce the concept of an n-skid product of algebras, for every n 6 ω. And we do it by generalizing, in a natural way, the concept of a factor pair of congruences. 1 What we obtain is a (potentially infinite) chain of degrees of indecomposability, which lie between subdirect irreducibility and direct indecomposability. In this talk we present this ongoing research, and the motivating examples. We will explain the different ways we have approached the concept and the results obtained so far. We focus our attention to particular varieties such as the variety of bounded meet semilattices and the varieties with compact unit congruences. Completions of Partial Information Systems James B Hart Middle Tennessee State University, U.S.A. [email protected] Partial information systems extend the classical notion of information system by removing the requirement that all tokens be consistent and that consistent sets entail all their subsets. Beginning with a partial information system S, we create the ideal completion PBC(S) in a way analogous to the well-known domain construction for information systems. The partial information system PS(PBC(S)) corresponding to this ideal completion can be understood as a “completion” of the system S in the following ways: (1) Up to equivalence, the consistency predicate of S may be order embedded in the consistency predicate of PS(PBC(S)) (2) The consistency predicate of S is strongly order-dense in the consistency predicate of PS(PBC(S)), and (3) Every pre-order homomorphism between the consistency predicates of partial information systems S and T extends to a pre-order homomorphism between the consistency predicates of PS(PBC(S)) and PS(PBC(T )). While exploring the relationship between S and PS(PBC(S)), motivation for partial information systems will be provided and variations on a number of concepts from the realm of denotational semantics will be introduced, including bounded-complete domains, Scott-continuous functions, and approximable relations. 2 Kites Tomasz Kowalski La Trobe University, Australia [email protected] (Joint work with Anatolij Dvurečenskij) Generalising a construction of Jipsen and Montagna, we produce certain special GBL-algebras (in fact, pseudo BL-algebras) we call kites. Intuitively, a kite consists of a power of the negative cone of an `-group G, say, (G− )I sitting on top of a power of the positive cone of G, say, (G+ )J . A number of well-known classes of algebras (for example, Boolean algebras, product logic algebras, Chang chain and its subdirect powers, and GMV algebras arising as intervals in Scrimger `-groups) can be viewed as kites. I will present some results on the structure of kites and the variety generated by them. In particular, I will show that subdirectly irreducible kites fall into five natural classes, and that the variety of kites is generated by finitely dimensional kites: such that the sets of indices I and J are finite. On Varieties of Lattice-ordered Effect Algebras Jan Kühr Palachý University in Olomouc, Czech Republic [email protected] Effect algebras are partially ordered partial Abelian monoids related to logical foundations of quantum mechanics. The class of lattice-ordered effect algebras is naturally equivalent to a certain variety E of algebras A = hA, ⊕, ¬, 0, 1i containing both the variety of MV-algebras and the variety (equivalent to the variety) of orthomodular lattices. Relative to E, MV-algebras are axiomatized by x⊕y ≈ y⊕x, and orthomodular lattices by x ⊕ x ≈ x. In the talk, we will focus on some subvarieties of E generated by horizontal sums of MV-algebras. Here, given a family {Ai : i ∈ I} of MV-algebras such L that Ai ∩ Aj = {0, 1} for all i 6= j, the horizontal sum of the Ai ’s is the algebra i∈I Ai 3 S with domain i∈I Ai , where (i) ¬a = ¬i a and (ii) a ⊕ b = a ⊕i b if a, b ∈ Ai for some i, and a ⊕ b = b otherwise. The Failure of Amalgamation Property for Semilinear Varieties of Residuated Lattices Antonio Ledda University of Cagliari, Italy [email protected] (Joint work with José Gil-Férez and Constantine Tsinakis) The word “amalgamation” refers to the process of combining a pair of algebras in such a way as to preserve a common subalgebra. This is made precise in the following definitions. Let K be a class of algebras of the same signature. A V-formation in K is a quintuple (A, B, C, i, j) where A, B, C ∈ K and i, j are embeddings of A into B, C, respectively. Given a V -formation (A, B, C, i, j) in K, (D, h, k) is said to be an amalgam of (A, B, C, i, j) in K if D ∈ K and h, k are embeddings of B, C, respectively, into D such that the compositions hi and kj coincide: j A ( C ( 6 k ( 6 i ( B 6 6 D h K has the amalgamation property (AP) if each V -formation in K has an amalgam in K. Amalgamations were first considered for groups by Schreier [14] in the form of amalgamated free products. The general form of the AP was first formulated by Fraïsse [3], and the significance of this property to the study of algebraic systems was further demonstrated in Jónsson’s pioneering work on the topic [4, 5, 6, 7, 8]. The added interest in the AP for algebras of logic is due to its relationship with various syntactic interpolation properties. We refer the reader to [10] for relevant references and an extensive discussion of these relationships; see also [11] and [9]. 4 There are no results to date of non-commutative varieties of residuated lattices enjoying the AP. The variety SemRL of semilinear (representable) residuated lattices, i.e., the variety generated by all totally ordered residuated lattices, seems like a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the AP are semilinear. An indication that this may not be the case comes from the fact that the variety RepLG of representable lattice-ordered groups fails the AP. Indeed, we prove that both SemRL and the variety SemCanRL of semilinear cancellative residuated lattices fail the AP. In addition, we prove that the much larger variety U of all residuated lattices with distributive lattice reduct and satisfying the identity x(y ∧ z)w ≈ xyw ∧ xzw also fails the AP. In fact, we show that any subvariety of this variety fails the AP, as long as its intersection with the variety of lattice-ordered groups fails the AP. There are two key ingredients in the proofs of these results. First, the fact that the specific V -formations that demonstrate the failure of the AP for the variety RepLG of representable lattice-ordered groups ([2, 13]; see [2, Theorem B]) also demonstrate its failure for SemRL and SemCanRL. The second key element in the proofs is the fact that each algebra in these varieties has a representation in terms of residuated maps of a chain [1, 12]. Bibliography [1] M. Anderson and C.C. Edwards. A representation theorem for distributive `-monoids. Canad. Math. Bull., 27:238–240, 1984. [2] V.V. Bludov and A.M.W. Glass. Amalgamation bases for the class of lattice-ordered groups. preprint, 2013. [3] R. Fraïsse. Sur l’extension aux relations de quelques proprietes des ordres. Ann. Sci. Éc. Norm. Supér, 71:363–388, 1954. [4] B. Jónsson. Universal relational structures. Math. Scand., 4:193–208, 1956. [5] B. Jónsson. Homomgeneous universal relational structures. Math. Scand., 8:137–142, 1960. [6] B. Jónsson. Sublattices of a free lattice. Canadian J. Math., 13:146–157, 1961. [7] B. Jónsson. Algebraic extensions of relational systems. Math. Scand., 11:179–205, 1962. [8] B. Jónsson. Extensions of relational structures. Proc. International Symposium on the Theory of Models, Berkeley, pages 146–157, 1965. 5 [9] E. Marchioni and G. Metcalfe. Craig interpolation for semilinear substructural logics. Mathematical Logic Quaterly, 58(6):468–481, 2012. [10] G. Metcalfe, F. Montagna, and C. Tsinakis. Amalgamation and interpolation in ordered algebras. submitted for publication, 2012. [11] G. Metcalfe, F. Paoli, and C Tsinakis. Ordered algebras and logic. Uncertainty and Rationality, Hosni, H., Montagna, F., Editors, Publications of the Scuola Normale Superiore di Pisa, Vol. 10, pages 1–85, 2010. [12] F. Paoli and C. Tsinakis. On Birkhoff’s common abstraction problem. Studia Logica, 100:1079–1105, 2012. [13] W. Powell and C. Tsinakis. The failure of the amalgamation property for varieties of representable `-groups. Math. Proc. Camb. Phil. Soc., 106:439–443, 1989. [14] O. Schreier. Die untergruppen der freien gruppen. Abh. Math. Sem. Univ. Hambur, 5:161–183, 1927. A Categorical Equivalence for Product Algebras Franco Montagna University of Siena, Italy [email protected] To be determined. Towards Multiset Consequence Relations Francesco Paoli University of Cagliari, Italy [email protected] (Joint work with Petr Cintula and José Gil-Férez) According to the dominant paradigm in Abstract Algebraic Logic (AAL), a (singleconclusion) consequence relation is a relation between a set of formulas and a 6 formula of a given language. The theory of such consequence relations is welldeveloped and well-rehearsed, see e.g. [3] or [4], and encompasses several different nonclassical logics, including many-valued and substructural logics. It can be argued, however, that the official AAL notion of consequence relation does not do justice to some of the most subtle and intriguing features of these logics, for which a multiset-theoretical notion of consequence relation would seem more appropriate. Unfortunately, apart from a few pioneering studies – see, for one, [1] – not much has been done on this count. The aim of this talk is reporting on some preliminary investigations we carried out on multiset consequence relations. In particular: • We introduce both a single-conclusion and a multiple-conclusion notion of multiset consequence relation, as well as a related concept of multiset consequence operator. We show that there is an order isomorphism between the complete lattices of multiset consequence operators and of multipleconclusion multiset consequence relations. • We study a notion of theory generated by a multiset of formulas. • We define multiset Hilbert systems, and prove for them an analogue of the Łoś-Suszko theorem for standard Hilbert systems. • Finally, we introduce a matrix semantics centred on a notion of fuzzy matrix, an algebra together with a fuzzy subset of designated values. As a case study, we prove that the multiset-theoretical companion of infinite-valued Łukasiewicz logic is sound and complete w.r.t. the class of all fuzzy matrices on the standard [0, 1] MV algebra whose set of designated values is a strictly monotonic function that preserves the Łukasiewicz T-norm. We are currently trying to circumscribe a workable definition of algebraisability and to attain a more abstract treatment of the matter in the style of Blok and Jónsson ([2], [5]). Bibliography [1] Avron A., “Simple consequence relations”, Information and Computation, 92, 1991, pp. 105-139. [2] Blok W.J., Jónsson B., “Equivalence of consequence operations”, Studia Logica, 83, 1-3, 2006, pp. 91–110. [3] Blok W.J., Pigozzi D., Algebraizable Logics, Memoirs of the AMS, Providence, RI, 1989. 7 [4] Font J.M., Jansana R., Pigozzi D., “A survey of abstract algebraic logic”, Studia Logica, 74, 2003, pp. 13-97. [5] Galatos N., Tsinakis C., “Equivalence of consequence relations: an order-theoretic and categorical perspective”, Journal of Symbolic Logic, 74, 3, 2009, pp. 780-810. Canonical Formulas and Residuated Lattices Luca Spada University of Salerno, Italy University of Amsterdam, Netherlands [email protected] Canonical formulas were introduced by Zakharyaschev as means to afford an axiomatisation of all subvarieties of Heyting algebras. In this talk, I will report on an ongoing research, jointly with N. Bezhanishvli, on a similar approach to the study of subvarieties of residuated lattices. Heyting Algebras as Intervals of Commutative, Cancellative Residuated Lattices William Young Huntington University, U.S.A. [email protected] For a commutative residuated lattice hA, ∧, ∨, ·, →, ei and a negative element a ≤ e of A, the interval [a, e] can be equipped with the structure of a residuated lattice in the following way: h[a, e], ∧, ∨, ◦, →e , ei, where for x, y ∈ [a, e], x◦y = (x·y)∨a and x →e y = (x → y) ∧ e. We observe that this construction is actually a conuclear image (σ(x) = x ∧ e) followed by a nuclear image (γ(x) = x ∨ a). It is well-known that M V -algebras are precisely the intervals of Abelian `-groups. The purpose of this talk is to begin the investigation of the class of residuated lattices that are the intervals of commutative, cancellative residuated lattices. Clearly, 8 these intervals must be bounded, integral, commutative residuated lattices, but are any other conditions necessary for a residuated lattice to be an interval of a commutative, cancellative residuated lattice? We use results of Montagna and Tsinakis, as well as the Mundici equivalence, to show the following result. Theorem 1. Let A be a bounded, integral, commutative residuated lattice. Then, A is a conuclear image of an M V -algebra iff A is an interval of a commutative, cancellative residuated lattice. Since Heyting algebras are the conuclear images of Boolean algebras (which are M V -algebras), we get as a corollary to the previous theorem that every Heyting algebra is an interval of a commutative, cancellative residuated lattice. Stone Duality for R0 -algebras (Nilpotent Minimum Algebras) with Internal States Hongjun Zhou Shaanxi Normal University, China [email protected] Nilpotent minimum algebras (briefly, NM-algebras) are the equivalent algebraic semantics of a fuzzy logic where the strong conjunction is modeled by the nilpotent minimum t-norm, a logic also independently introduced by Guojun Wang in the mid 1990s. In this paper, we first establish a Stone duality for the category of MV-skeletons of NM-algebras and the category of three-valued Stone spaces. Then we extend Flaminio-Montagna internal states to NM-algebras. Such internal states turn out to be idempotent MV-endomorphisms of NM-algebras. Lastly we present a Stone duality for the category of MV-skeletons of NM-algebras with a FlaminioMontagna internal state and the category of three-valued Stone spaces with a Zadeh type idempotent continuous function. 9

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