Injective and projective semimodules over involutive semirings

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [Formula: see text] to be a subalgebra of an involutive residuated lattice, where [Formula: see text] is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

Automatic Inference of Term Equivalence in Term Rewriting Systems

In this paper we propose a parametric technique to automatically infer algebraic property-oriented specifications from Term Rewriting Systems. Namely, given the source code of a TRS we infer a specification which consists of a set of most general equations relating terms that rewrite, for all possible instantiations, to the same set of normal forms.The semantic-based inference method that we propose can cope with non-constructor-based TRSs, and considers non-ground terms. Particular emphasis is posed to avoid the generation of “redundant” equations that can be a logical consequence of other ones.To experiment on the validity of our proposal we have considered an instance of the method employing a novel (condensed) semantics for left-linear TRSs and we have implemented a “proof of concept” prototype in Haskell which is available online.

On the concept of term equivalence

The article considers some terminological aspects in the process of harmonization of legislation reflecting on different approaches to the study of terms and especially to synonymy and term equivalence. The various mechanisms available to the translator are examined within the EU context and against the background of Bulgaria’s legal culture. The analysis is based on translations of EU legislation from English into Bulgarian and highlights felicitous choices and techniques employed, as well as recurring inconsistencies in the long and arduous process of approximation of legislation.

THE PERKINS SEMIGROUP HAS CO-NP-COMPLETE TERM-EQUIVALENCE PROBLEM

A semigroup term is a finite word in the alphabet x1, x2,…. The length of a term p, denoted by |p|, is the number of variables in p, including multiplicities. The term-equivalence problem for a finite semigroup S has as an instance a pair of terms {p,q} with size |p| + |q| and asks whether p ≈ q is an identity over S. It is proved here that [Formula: see text], the six-element Perkins semigroup, has co-NP-complete term-equivalence problem, a result which leads to the completion of the classification of he term-equivalence problems for monoid extensions of aperiodic Rees matrix semigroups. From the main result it follows that there exist finite semigroups with tractable term-equivalence problems but having subsemigroups and homomorphic images with co-NP-complete term-equivalence problems.