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 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.     

Completion and finite embeddability property for residuated ordered algebras.

A residuated ordered algebra is a partially ordered set with additional ‘residuated’ operations. A construction is presented that, from any partial subalgebra of a residuated ordered algebra, constructs a complete algebra into which the partial subalgebra embeds. Conditions are given under which the constructed algebra is finite whenever a finite partial subalgebra is chosen. This implies the ‘finite embeddability property’ for the given class of residuated ordered algebras. In the case that the whole algebra is chosen as the partial subalgebra, the construction is a completion of the underlying order of the algebra. A scheme of inequalities is described that are shown to have the property of being preserved by the above construction. These preservation results thus extend the results on the finite embeddability property and completion.     

On distributive residuated groupoids

In this note we extend the well-known fact (see [1], p. 45) that an implicative (or Brouwerian) lattice is distributive to a large class of residuated groupoids.     

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.     

Automorphism groups of finite groupoids

Given a finite set M of size n and a subgroup G of Sym(M), G is pertinent iff it is the automorphism group of some groupoid ??M; *??. We examine when subgroups of Sym(M) are and are not pertinent. For instance, A _n , the alternating group on M, is not pertinent for n > 4. We close by indicating a natural extension of our ideas, which relates to a question of M. Gould.     

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 finite cutset property

A cutset of H is a subset of ∪ H which meets every element of H. H has the finite cutset property if every cutset of H contains a finite one. We study this notion, and in particular how it is related to the compactness of H for the natural topology. MSC: 04A20, 54D30.     

The Rohlin property for shifts of finite type

We show that an automorphism of a unital AF C^*-algebra with a certain approximate Rohlin property has the Rohlin property. This generalizes a result of Kishimoto. Using this we show that the shift automorphism on the bilateral C^*-algebra associated with an aperiodic irreducible shift of finite type has the Rohlin property.     

The property O n for finite seminormal functors

We give a finite combinatorial test for finite seminormal functors to possess the property O _n and use it in establishing that in some cases this property leads to some well-known functors. For example, if some functor F possesses the property O ₂ then F ₂ coincides with either exp₂ or the squaring functor. Hence we conclude that if F(D ^ω ₁) and D ^ω ₁ are homeomorphic then F ₂ is either exp₂ or (·)².     

The finite model property for semilinear substructural logics

In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.     

The E-theoretic descent functor for groupoids

The paper establishes, for a wide class of locally compact groupoids Γ, the E-theoretic descent functor at the C^∗-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. Section 2 shows that Γ-actions on a C₀(X)-algebra B, where X is the unit space of Γ, can be usefully formulated in terms of an action on the associated bundle B^?. Section 3 shows that the functor B→C^∗(Γ,B) is continuous and exact, and uses the disintegration theory of J. Renault. Section 4 establishes the existence of the descent functor under a very mild condition on Γ, the main technical difficulty involved being that of finding a Γ-algebra that plays the role of Cbcont(T,B) in the group case.