On the finite embeddability property for residuated ordered groupoids

The finite embeddability property (FEP) for integral, commutative residuated ordered monoids was established by W. J. Blok and C. J. van Alten in 2002. Using Higman's finite basis theorem for divisibility orders we prove that the assumptions of commutativity and associativity are not required: the classes of integral residuated ordered monoids and integral residuated ordered groupoids have the FEP as well. The same holds for their respective subclasses of (bounded) (semi-)lattice ordered structures. The assumption of integrality cannot be dropped in general--the class of commutative, residuated, lattice ordered monoids does not have the FEP--but the class of -potent commutative residuated lattice ordered monoids does have the FEP, for any .

     

The finite embeddability property for residuated groupoids

A very simple proof of the finite embeddability property for residuated distributive-lattice-ordered groupoids and some related classes of structures is presented. In particular, this gives an answer to the question, posed by Blok and van Alten, whether the class of residuated ordered groupoids has the property. The presented construction improves the computational-complexity upper bound of the universal theory of residuated distributive-lattice-ordered groupoids given by Buszkowski and Farulewski; for chains in the class, a tight bound is obtained.     

The finite embeddability property for residuated lattices, pocrims and BCK-algebras

A class of algebras has the finite embeddability property (FEP) if every finite partial subalgebra of an algebra in the class can be embedded into a finite algebra in the class. We investigate the relationship of the FEP with the finite model property (FMP) and strong finite model property (SFMP).? For quasivarieties the FEP and the SFMP are equivalent, and for quasivarieties with equationally definable principal relative congruences the three notions FEP, FMP and SFMP are equivalent. The variety of intuitionistic linear algebras –which is known to have the FMP–fails to have the FEP, and hence the SFMP as well. The variety of integral intuitionistic linear algebras (also known as the variety of residuated lattices) does possess the FEP, and hence also the SFMP. Similarly contrasting statements hold for various subreduct classes. In particular, the quasivarieties of pocrims and of BCK-algebras possess the FEP. As a consequence, the universal theories of the classes of residuated lattices, pocrims and BCK-algebras are decidable. Received February 16, 2001; accepted in final form November 2, 2001. RID="h1" ID="h1"The second author was supported by a postdoctoral research fellowship of the National Research Foundation of South Africa, hosted by the University of Illinois at Chicago.     

Semilinear substructural logics with the finite embeddability property

Three semilinear substructural logics \({\mathbf{HpsUL}}_\omega ^*\), \({\mathbf{UL}}_\omega \) and \({\mathbf{IUL}}_\omega \) are constructed. Then the completeness of \({ \mathbf{UL}}_\omega \) and \({\mathbf{IUL}}_\omega \) with respect to classes of finite UL and IUL-algebras, respectively, is proved. Algebraically, non-integral \({\mathbf{UL}}_\omega \) and \({\mathbf{IUL}}_\omega \)-algebras have the finite embeddability property, which gives a characterization for finite UL and IUL-algebras.     

Decomposability and embeddability of discretely Henselian division algebras

LetF be a discretely Henselian field of rank one, with residue fieldk a number field, and letD/F be anF-division algebra. We conduct an exhaustive study of the decomposability of an arbitraryD. Specifically, we prove the following:D has a semiramified (SR)F-division subalgebra if and only ifD has a totally ramified (TR) subfield. However, there may be TR subfields not contained in any SR subalgebra. IfD has prime-power index, thenD is decomposable if and only ifD properly contains a SR division subalgebra. Equivalently,D has a decomposable Sylow factor if and only if ii(D ^⊗n )≠1/n i(D) for somen dividing the period ofD, that is, if and only if the index fails to mimic the behavior of the period ofD. There exists indecomposableD with prime-power periodp ² and indexp ³. Every proper division subalgebra ofD is indecomposable. Conversely, every indecomposableF-division algebra ofp-power index embeds properly in someD ofp-power index if and only ifk does not have a certain strengthened form of class field theory’s Special Case. Semiramified division algebras and division algebras of odd index always properly embed. Finally, these results apply to an extent overk(t), and we prove that there exist indecomposablek(t)-division algebras of periodp ² and indexp ³, solving an open problem of Saltman. Dedicated to the memory of Amitsur Research supported in part by NSF Grant DMS-9100148.     

Distributive residuated frames and generalized bunched implication algebras

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.     

On strong embeddability and finite decomposition complexity

The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, we study the permanence properties of strong embeddability for metric spaces. We show that strong embeddability is coarsely invariant and it is closed under taking subspaces, direct products, direct limits and finite unions. Furthermore, we show that a metric space is strongly embeddable if and only if it has weak finite decomposition complexity with respect to strong embeddability.     

The ordered field property and a finite algorithm for the Nash bargaining solution

This note proves that the two person Nash bargaining theory with polyhedral bargaining regions needs only an ordered field (which always includes the rational number field) as its scalar field. The existence of the Nash bargaining solution is the main part of this result and the axiomatic characterization can be proved in the standard way with slight modifications. We prove the existence by giving a finite algorithm to calculate the Nash solution for a polyhedral bargaining problem, whose speed is of orderBm(m-1) (m is the number of extreme points andB is determined by the extreme points).     

Injective hulls for ordered algebras

We study the injectivity of ordered algebras with respect to the class of homomorphisms that are order-embeddings and one more class of morphisms. We do this in the category where morphisms need not be homomorphisms, but satisfy a condition which is weaker than operation-preservation. In this setting, the injective objects turn out to be precisely sup-algebras. We also show how to construct injective hulls of ordered algebras satisfying certain conditions.     

R 0-algebras and weak dually residuated lattice ordered semigroups

We introduce the notion of weak dually residuated lattice ordered semi-groups (WDRL-semigroups) and investigate the relation between R ₀-algebras and WDRL-semigroups. We prove that the category of R ₀-algebras is equivalent to the category of some bounded WDRL-semigroups. Moreover, the connection between WDRL-semigroups and DRL-semigroups is studied.     

Interior and closure operators on bounded residuated lattice ordered monoids

GMV-algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior GMV-algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on DRl-monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on GMV-algebras.     

Cylindric algebras and finite polyadic algebras

In this survey paper the short history of cylindric and finitary polyadic algebras (term-definitionally equivalent to quasi-polyadic algebras) is sketched, and the two concepts are compared. Roughly speaking, finitary polyadic algebras constitute a subclass of cylindric algebras that include a transposition operator being strong enough. We discuss the following question: should the definition of cylindric algebras include a transposition operator? Results confirm that the existence of a transposition operator ensures representability (by relativised set algebras). The different variants of cylindric algebras including a transposition operator play an important role in the theory of cylindric-like algebras.     

Decision methods for linearly ordered Heyting algebras

The decision problem for positively quantified formulae in the theory of linearly ordered Heyting algebras is known, as a special case of work of Kreisel, to be solvable; a simple solution is here presented, inspired by related ideas in Gödel-Dummett logic.