Proof Theory and Ordered Groups

Ordering theorems, characterizing when partial orders of a group extend to total orders, are used to generate hypersequent calculi for varieties of lattice-ordered groups (l-groups). These calculi are then used to provide new proofs of theorems arising in the theory of ordered groups. More precisely: an analytic calculus for abelian l-groups is generated using an ordering theorem for abelian groups; a calculus is generated for l-groups and new decidability proofs are obtained for the equational theory of this variety and extending finite subsets of free groups to right orders; and a calculus for representable l-groups is generated and a new proof is obtained that free groups are orderable

Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics

We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for L. These include: hypersequent calculi for A and L and terminating versions of these calculi; labelled single sequent calculi for A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur

Algebraic Analysis of some Classes of Fuzzy Ordered Structures

Neka je A neprazan skup  i ℒ = (L, ≤) proizvoljna mreža sa nulom i jedinicom. Svako preslikavanje µ: A → L zovemo rasplinuti podskup od A. U ovoj tezi proučavali smo rasplinute posete i relacije rasplinutog poretka. Uveli smo neke nove pojmove: rasplinuta uređena grupa, rasplinuti pozitivan konus, rasplinuti negativan konus, rasplinuta mrežno uređena grupa. Posmatrajući strukturu svih relacija slabog rasplinutog poretka koje su podskup klasične relacije poretka ≤ , došli smo do zaključka da ova struktura predstavlja kompletnu mrežu. Takođe, važan zadatak je bio da ispitamo egzistenciju rasplinute mrežno uređene podgrupe l –uređene grupe koja nije linearno uređena. Bitan rezultat je rasplinuta mrežno uređena podgrupa date mrežno uređene grupe G, koja je konstruisana pomoću mreže svih kompleksnih l –podgrupa od G.Let A be a nonempty set, and let ℒ = (L, ≤) be a lattice with 0 and 1. The mapping: µ: A → L is called a fuzzy subset of A. In this work we investigated fuzzy posets and fuzzy ordering relations. We introduced some new notions: fuzzy ordered groups, fuzzy positive cone, fuzzy negative cone, fuzzy lattice ordered group. Considering a structure of all weak fuzzy orderings contained in the crisp order ≤, we concluded that this structure represents a complete lattice. Also, an important task was to investigate the existence of a fuzzy lattice ordered subgroup of an l–ordered group which is not linearly ordered. A main result is a fuzzy lattice ordered subgroup of a given lattice ordered group G, which is constructed by the lattice of all convex l-subgroups of G

SPECIAL VALUED l-GROUPS

Special elements and values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call such l-groups special valued. In this paper, we first show that several familiar structures of l-groups, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity; that is, knowing which element of the lattice is the group identity, we can pick out in the lattice all the dements of polars, minimal primes, and the lex kernel. This then leads to an easy proof that special elements can be recognized from the lattice and the identity. We then prove several results about the class S of special-valued l-groups. We give a simple and direct proof that S is closed with respect to joins of convex l-subgroups, incidentally giving a direct proof that S is a quasi torsion class. This proof is then used to show that the special-valued and finite-valued kernels of l-groups are recognizable from the lattice and the identity. We show also that the lateral completion of a special-valued l-group is special-valued and is an a*-extension of the original l-group. Our most important result is that the lateral completion of a completely-distributive normal-valued l-group is special-valued. This lends itself easily to a new and similar proof of Ball, Conrad, and Darnel's result that every normal-valued l-group can be l-embedded into a special-valued l-group. Readers familiar with the impact of the Conrad-Harvey-Holland Theorem on abelian l-groups will recognize the importance of the last theorem to the study of the class of normal-valued l-groups and to the study of proper varieties of l-groups, all of which are normal valued