My primary
research interests lie in the areas of Universal Algebra, Ordered
Algebraic Structures and Algebraic Logic. In particular, the
algebraic structure of residuated lattices has been central in my
research. Residuated lattices generalize various well-studied structures and
have applications and connections to different areas of mathematics.
Residuated lattices generalize significant, well-studied and disparate algebraic structures, including lattice-ordered groups and Brouwerian algebras, and combine two basic algebraic systems - lattices and monoids. Hence, they constitute a unique platform for the unified treatment of seemingly unrelated structures. They were originally constructed as abstractions of lattices of ideals of rings, and they encompass the categorical notion of adjunction, implemented in the setting of partially ordered sets. Despite its generality, the theory of residuated lattices is robust and provides tools and models applicable to the particular specializations. It is interesting that residuated lattices, structures with prominent algebraic content and history, turn out to constitute algebraic semantics for substructural logics, a fact that establishes a bi-directional connection between logic and algebra. Additionally, residuated structures arise naturally in theoretical computer science, in the context of programming languages development.
The main focus of my research has been the study of the algebraic properties of the variety of residuated lattices and its subvarieties, as well as the investigation of the connections of residuated lattices to mathematical logic.
In particular, in [1], I explore algorithmic questions about the variety of distributive residuated lattices. A variety is said to have an undecidable word problem, if it contains an algebra such that there is no algorithm that decides whether or not any two given words in the absolutely free algebra of the variety represent the same element of the given algebra. It is well known that the word problem for the varieties of semigroups, groups and lattice-ordered groups is undecidable. The main result of this paper is the undecidability of the word problem for a range of varieties including the variety of distributive residuated lattices and the variety of commutative distributive residuated lattices. The proof makes use of the concept of an n-frame, introduced by von Neumann. The origins of the technique and the intuition behind it are completely geometric.
In [2], I establish a correspondence between positive universal formulas and varieties of residuated lattices. Given a positive universal formula in the language of residuated lattices, I construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. This allows one to write down bases of equations for the varieties generated by positive universal classes. As an example, axiomatizations for representable residuated lattices and for ones that are the union of their cones can be derived. The main result is further applied to show that the join of two finitely based commutative subvarieties of residuated lattices is also finitely based.
Structure theory problems are solved in my joint paper [3]. Cancellative residuated lattices are always infinite and they form a variety. A detailed analysis of their behavior is presented and they are compared to lattice-ordered groups. Among other things, it is shown that cancellative residuated lattices do not satisfy any non-trivial lattice identity. In particular, the lattice of subvarieties of lattices can be order-embedded to the subvariety lattice of cancellative residuated lattices. Additionally, a bijective correspondence between the varieties of lattice-ordered groups and the varieties of their negative cones is established. This correspondence, which is a lattice isomorphism, is further enhanced by a translation of the equational bases of the corresponding varieties.
In [4], we describe two constructions that provide embeddings of residuated lattices into involutive ones. Both constructions preserve the properties of commutativity and distributivity. In conjunction with results in [1], we obtain the undecidability of the quasi-equational theory of involutive, commutative, distributive residuated lattices. Moreover, we prove that the equational theory of these residuated lattices is decidable, obtaining, in this way, a clear understanding of the separation between the two theories. As a consequence, we obtain decision results for systems of relevance logic.
The atoms of the lattice of subvarieties of residuated lattices are investigated in [5]. A continuum of atomic subvarieties is constructed, all of which satisfy the idempotent law for multiplication. These varieties are generated by totally ordered algebras on the same underlying chain and have a conservative multiplication. It is shown that the generating algebras are in a bijective correspondence to the mechanical bi-infinite words with irrational slope. Moreover, it is shown that there are only two cancellative atomic varieties of residuated lattices and only two commutative idempotent ones.
In [6], standard MV-algebras – algebraic models of Lukasiewicz’s Multi-Valued logic – are generalized to a class that includes lattice-ordered groups and generalized Boolean algebras. It is shown that every generalized MV-algebra is a part of a lattice-ordered group and that it decomposes into the Cartesian product of an integral generalized MV-algebra and a lattice-ordered group. The decidability of the equational theory of the variety is an easy corollary of the representation theorems in the paper. It is known that the correspondence between MV-algebras and intervals of Abelian lattice-ordered groups can be extended to a categorical equivalence. We expand this result to generalized MV-algebras and nuclei retractions on arbitrary lattice-ordered groups.
The work in [7] and [8] is an algebraic treatment of substructural logics over the full Lambek calculus. In [7], we note that pointed residuated lattices are algebraic semantics for these logics, discuss the generation of congruences and congruence filters, obtain a deduction theorem and explore the connections between the properties of interpolation and amalgamation. Having these results as a basis, in [8] we present a vast generalization of Glivenko’s theorem. In particular, we show that the blocks of the Glivenko equivalence among subvarieties of residuated lattices are intervals and provide graded conditions of involutiveness for subvarieties relative to which various types of Glivenko theorems hold. Furthermore, we extend known translations between logical systems to the setting of substructural logic.
In [9], I present a duality theory for the category of bounded distributive residuated lattices, with bounded homomomorphisms, that is an extension of Priestley duality. The strict connection of the normal operators of multiplication and the division operations allow the introduction of a unique ternary relation on the dual topological spaces, along with the order relation. The results can be applied in the study of the amalgamation property for classes of distributive residuated lattices.
The study of non-associative
residuated lattices is undertaken in [10]. In particular, we present a Hilbert
system with algebraic semantics the variety of residuated lattice-ordered
groupoids with unit. The system is shown to enjoy the strong separation
property, a fact that allows for the axiomatization of the classes of
subreducts of the main variety. Additionally, we construct a Gentzen system
that is equivalent to the original Hilbert system and has the cut-elimination
property, obtaining in this way the decidability of the equational theory.
Finally, the existence of the embedding that proves the finite embedability
property for certain subvarieties follows as a corollary of the main
result.
References:
[1] The undecidability of the word problem for distributive residuated lattices, Ordered Algebraic Structures (J. Martinez, ed.), Kluwer Academic Publishers, Dordrecht, 2002, 231-243.
[2] Equational bases for joins of residuated-lattice varieties, Studia Logica 76 (2004), no. 2, 227-240.
[3] Cancellative residuated lattices, with P. Bahls, J. Cole, P. Jipsen and C. Tsinakis, Algebra Universalis 50 (2003), no. 1, 83-106.
[4] Adding involution to residuated structures, with J. Raftery, forthcoming Studia Logica 77 (2004), no. 1, pp 27.
[5] Minimal varieties of residuated lattices, to appear in Algebra Universalis, 2003.
[6] Generalized MV-algebras, with C. Tsinakis, submitted, 2004.
[7] Substructural logics: algebraization and interpolation, with H. Ono, in preparation.
[8] Substructural logics: Glivenko theorem and other translations, with H. Ono, in preparation.
[9] A Priestley duality for bounded distributive residuated lattices, in preparation.
[10] Non-associative FL, with H. Ono, in preparation.