Density revisited

Metcalfe, George; Tsinakis, Constantine (2017). Density revisited. Soft computing, 21(1), pp. 175-189. Springer 10.1007/s00500-016-2420-7

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In this (part survey) paper,we revisit algebraic and proof-theoretic methods developed by Franco Montagna and his co-authors for proving that the chains (totally ordered members) of certain varieties of semilinear residuated lattices embed into dense chains of these varieties, a key step in establishing standard completeness results for fuzzy logics. Such “densifiable” varieties are precisely the varieties that are generated as quasivarieties by their dense chains.By showing that all dense chains satisfy a certain e-cyclicity equation, we give a short proof that the variety of all semilinear residuated lattices is not densifiable (first proved by Wang and Zhao). We then adapt the Jenei–Montagna standard completeness proof for monoidal t-norm logic to show that any variety of integral semilinear residuated lattices axiomatized by additional lattice-ordered monoid equations is densifiable. We also generalize known results to show that certain varieties of cancellative semilinear residuated lattices are densifiable. Finally, we revisit the Metcalfe–Montagna proof-theoretic approach, which establishes densifiability of a variety via the elimination of a density rule for a suitable hypersequent calculus, focussing on the case of commutative semilinear residuated lattices.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Metcalfe, George

Subjects:

500 Science > 510 Mathematics

ISSN:

1432-7643

Publisher:

Springer

Language:

English

Submitter:

George Metcalfe

Date Deposited:

26 Apr 2017 12:36

Last Modified:

03 Nov 2017 02:30

Publisher DOI:

10.1007/s00500-016-2420-7

BORIS DOI:

10.7892/boris.92780

URI:

https://boris.unibe.ch/id/eprint/92780

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