Constantine Tsinakis

A Survey of Residuated Lattices

Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of lattice-ordered groups, ideal lattices of rings, linear logic and multi-valued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability.

Minimal Varieties of Involutive Residuated Lattices

We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.

Perfect GMV-Algebras

The focus of this article is the class of perfect GMV-algebras, which includes all noncommutative analogs of perfect MV-algebras. As in the commutative case, we show that each perfect GMV-algebra possesses a single negation, it is generated by its infinitesimal elements, and can be uniquely realized as an interval in a lexicographical product of the lattice-ordered group of integers and an arbitrary lattice-ordered group. Further, we establish that the category of perfect GMV-algebras is equivalent to the category of all lattice-ordered groups. The variety of GMV-algebras generated by the class of perfect GMV-algebras plays a key role in our considerations. Among other results, we describe a finite equational basis for this variety and prove that it fails to satisfy the amalgamation property. In fact, we show that uncountably many of its subvarieties fail this property.

Free Products in Varieties of Lattice-Ordered Groups

The concept of a free product is fundamental to the study of any kind of algebraic system. Intuitively, a free product takes a family of algebras from a given class and combines it in the “loosest” or “freest” way possible. By this it is meant that any other algebra generated by the given family must be a homomorphic image of the free product.

Products of Classes of Residuated Structures

The central result of this paper provides a simple equational basis for the join, IRL∨LG, of the variety LG of lattice-ordered groups (ℓ-groups) and the variety IRL of integral residuated lattices. It follows from known facts in universal algebra that IRL∨LG=IRL×LG. In the process of deriving our result, we will obtain simple axiomatic bases for other products of classes of residuated structures, including the class IRL×sLG, consisting of all semi-direct products of members of IRL by members of LG. We conclude the paper by presenting a general method for constructing such semi-direct products, including wreath products.

The finite basis theorem for relatively normal lattices

This study has as its primary objective to provide an in depth analysis of the structure of relatively normal lattices. A featured result is a purely lattice-theoretic generalization of ConradsFinite Basis Theorem for lattice-ordered groups.

Globally determined lattices and semilattices

whenever f is an n-ary operation in F and A1,.. . , A n are complexes of A. A class Yg of algebras is said to be globally determined if any two members of having isomorphic globals must themselves be isomorphic. Tamura and Shafer [9] noted that the class of all groups is globally determined, and from the easy proof of this fact follows the corresponding result for rings. Certain classes of semigroups, including the class of all finite semilattices, were shown by Gould and Iskra [2] to be globally determined. Such results were established for various classes of semigroups by Tamura [8], Tamura and Sharer [10], Va~enin [11], and Mogiljanskaja [3] [4] [5], who also exhibited [6] [7] pairs of non-isomorphic infinite semigroups having isomorphic globals. The fact that finite mono-unary algebras are globally determined was established by Drgtpal [1], along with a counterexample for the infinite case. We shall show that the class of all semilattices with identity is globally determined (Theorem 1.3) and utilize this result to prove that the class of all lattices is globally determined (Theorem 2.2).

Gödel incompleteness in AF C*-algebras

Abstract For any (possibly, non-unital) AF C*-algebra A with comparability of projections, let D(A) be the Elliott partial monoid of A, and G(A) the dimension group of A with scale D(A). For D ⊆ D(A) a generating set of G(A) let 𝒫 be the set of all formal inequalities a 1 + ⋯ + ak ≤ b 1 + ⋯ + bl satisfied by G(A), for any ai, bj ∈ D. By Elliotts classification, 𝒫 together with the list of all sums a 1 + ⋯ + ak ∈ D(A) uniquely determines A. Can 𝒫 be Gödel incomplete, i.e., effectively enumerable but undecidable? We give a negative answer in case D is finite, and a positive answer in the infinite case. We also show that the range of the map A ↦ D(A) precisely consists of all countable partial abelian monoids satisfying the following three conditions: (i) a + b = a + c ⇒ b = c, (ii) a + b = 0 ⇒ a = b = 0 and (iii) ∀a, b ∈ E ∃c ∈ E such that either a + c = b or b + c = a.

DECOMPOSITIONS FOR RELATIVELY NORMAL LATTICES

Continuing the work begun in Snodgrass and Tsinakis (26, 27), we develop a family of decomposition theorems for classes of relatively normal lat- tices. These results subsume and are inspired by known decomposition theorems in order-algebra due to P. Conrad and D. B. McAlister. As direct corollaries of the main results, we obtain corresponding decomposition theorems for classes of partially ordered sets.

The distributive lattice free product as a sublattice of the abelian l -group free product

This paper establishes an important link between the class of abelian l -groups and the class of distributive lattices with a distinguished element. This is accomplished by describing the distributive lattice free product of a family of abelian l -groups as a naturally generated sublattice of their abelian l -group free product.