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Residuated lattices

Morgan Ward, R. P. Dilworth
1939 Transactions of the American Mathematical Society  
We prove a similar result for residuation. Theorem 8.2. A lattice built up out of a set of residuated lattices connected into a chain by nodes can be residuated.  ...  Let © ie a lattice in which the following conditions hold: N 1. The lattice © may be residuated. Then © ¿s a Noether lattice.  ...  We first consider the following lattice SBi; and we define a multiplication xy over Söi by the following  ... 
doi:10.1090/s0002-9947-1939-1501995-3 fatcat:4esiadto5nasdmvpb5tahs5uee

Residuated Lattices

M. Ward, R. P. Dilworth
1938 Proceedings of the National Academy of Sciences of the United States of America  
A lattice built up out of a set of residuated lattices connected into a chain by nodes can be residuated. THEOREM.  ...  A direct product of residuated lattices can be residuated; conversely if a residuated lattice can be expressed as a direct product, each of its factors can be residuated. THEOREM.  ... 
doi:10.1073/pnas.24.3.162 pmid:16588213 pmcid:PMC1077055 fatcat:3flgiz5eqbfzffummrig6wrlm4

Residuated Lattices

Morgan Ward, R. P. Dilworth
1939 Transactions of the American Mathematical Society  
We prove a similar result for residuation. Theorem 8.2. A lattice built up out of a set of residuated lattices connected into a chain by nodes can be residuated.  ...  Let © ie a lattice in which the following conditions hold: N 1. The lattice © may be residuated. Then © ¿s a Noether lattice.  ...  We first consider the following lattice SBi; and we define a multiplication xy over Söi by the following  ... 
doi:10.2307/1990008 fatcat:hubga6jamzfdbcwghndfpu7ddu

Archimedean Residuated Lattices

Dumitru Buşneag, Dana Piciu, Antoneta Jeflea
2010 Annals of the Alexandru Ioan Cuza University - Mathematics  
of Nachbin type for residuated lattices.  ...  For a residuated lattice A we denote by Ds(A) the lattice of all deductive systems (congruence filters) of A.  ...  of a residuated lattice.  ... 
doi:10.2478/v10157-010-0017-5 fatcat:ygpw6x3655a63au2c5zmbohcoe

Sectionally residuated lattices

I. Chajda
2005 Miskolc Mathematical Notes  
Residuated lattices were introduced by Ward and Dilworth [6] and studied by several authors. Two monographs contain a compendium on residuated lattices.  ...  Thus, considering ⊗ = ∧, N 5 is not a residual lattice.  ... 
doi:10.18514/mmn.2005.96 fatcat:gydtt6qfgvcgxkhpjkvoyacrja

Quasicomplemented residuated lattices [article]

Saeed Rasouli
2019 arXiv   pre-print
The notion of disjunctive residuated lattices is introduced and it is observed that a residuated lattice is Boolean if and only if it is disjunctive and quasicomplemented.  ...  In this paper, the class of quasicomplemented residuated lattices is introduced and investigated, as a subclass of residuated lattices in which any prime filter not containing any dense element is a minimal  ...  Let A be a residuated lattice.  ... 
arXiv:1904.10302v1 fatcat:gwzhdpqdyfenpozzhjnvytjqiy

Mp-residuated lattices [article]

Saeed Rasouli, Amin Dehghani
2022 arXiv   pre-print
This paper is devoted to the study of a fascinating class of residuated lattices, the so-called mp-residuated lattice, in which any prime filter contains a unique minimal prime filter.  ...  The class of mp-residuated lattices is characterized by means of pure filters.  ...  Let A be a residuated lattice.  ... 
arXiv:2203.15018v1 fatcat:z3uzhy6rynfhfn4pc4vdd4byfi

Cancellative residuated lattices

P. Bahls, J. Cole, N. Galatos, P. Jipsen, C. Tsinakis
2003 Algebra Universalis  
Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( -groups).  ...  In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice.  ...  For a survey of residuated lattices we refer to [21] . The class RL of all residuated lattices is easily seen to be a variety [7] .  ... 
doi:10.1007/s00012-003-1822-4 fatcat:clykxn6i75efzeblp5d4ushymq

Commutative idempotent residuated lattices

David Stanovský
2007 Czechoslovak Mathematical Journal  
(Note that, in fact, C 2 is the only two-element residuated lattice and C 3 is the only non-integral three-element residuated lattice.)  ...  A residuated lattice is called integral if all its elements are negative. Let C 2 be the two-element CI residuated lattice, C 2 = {0, 1}, e = 1.  ... 
doi:10.1007/s10587-007-0055-7 fatcat:dtn4xcbkl5clljgzdefulmekyy

Kites and Residuated Lattices [article]

Michal Botur, Anatolij Dvurečenskij
2017 arXiv   pre-print
We investigate a construction of an integral residuated lattice starting from an integral residuated lattice and two sets with an injective mapping from one set into the second one.  ...  We show that the variety of integral residuated lattices generated by kites is generated by all finite-dimensional kites. In particular, we describe some homomorphisms among kites.  ...  Kites Residuated Lattices We present a construction of kite residuated lattices starting from an integral residuated lattice.  ... 
arXiv:1707.00267v1 fatcat:i5fxcdnxyfglfiama4khe3vcpu

Non-Commutative Residuated Lattices [chapter]

R. P. Dilworth
1990 The Dilworth Theorems  
In Part I the properties of non-commutative multiplication and residuation over a lattice are developed.  ...  The second division of the paper deals with the structure of non-commutative residuated lattices in the vicinity of the unit element.  ...  Archimedean residuated lattices 8.  ... 
doi:10.1007/978-1-4899-3558-8_33 fatcat:vgdj6kcgkvbpvatyah4we5gt7a

Monadic Bounded Residuated Lattices

Jiří Rachůnek, Dana Šalounová
2011 Order  
Keywords Bounded integral residuated lattice · Monadic residuated lattice · Algebras of logics · Quantifier Mathematics Subject Classifications (2010) 03B50 · 06D20 · 06D35 · 06F05 Introduction Monadic  ...  Bounded integral residuated lattices form a large class of algebras which contains algebraic counterparts of several propositional logics behind many-valued reasoning and intuitionistic logic.  ...  We define functional monadic residuated lattices and show that every such algebra is a monadic residuated lattice.  ... 
doi:10.1007/s11083-011-9236-y fatcat:x4woubxx7favnjjpgoy54n763i

n-normal residuated lattices [article]

Saeed Rasouli, Michiro Kondo
2018 arXiv   pre-print
The notion of n-normal residuated lattice, as a class of residuated lattices in which every prime filter contains at most n minimal prime filters, is introduced and studied.  ...  Before that, the notion of ω-filter is introduced and it is observed that the set of ω-filters in a residuated lattice forms a distributive lattice on its own, which includes the set of coannulets as a  ...  Let A be a residuated lattice.  ... 
arXiv:1812.11511v1 fatcat:zg6djymcqnha3dfzwqy422gtq4

Conic Idempotent Residuated Lattices [article]

Wesley Fussner, Nick Galatos
2022 arXiv   pre-print
We show that every variety generated by conic idempotent residuated lattices enjoys the congruence extension property. In particular, this holds for semilinear idempotent residuated lattices.  ...  Using this, we show that although conic idempotent residuated lattices do not enjoy the amalgamation property, the natural class of rigid and conjunctive conic idempotent residuated lattices has the strong  ...  residuated lattices.  ... 
arXiv:2208.09724v1 fatcat:fz23ke7xqneepbhruu5pmb7tsm

Abstract Residuation over Lattices [chapter]

R. P. Dilworth
1990 The Dilworth Theorems  
RESIDÜATION OVER LATTICES* R. P. DILWORTH Introduction. The idea of residuation goes back to Dedekind [3 ], f who introduced it in the theory of modules.  ...  Hence we shall consider a residual defined over a system having only these two operations, that is, over a lattice [2] .  ...  LI, L2, L3, and L4 are his axioms for a lattice.  ... 
doi:10.1007/978-1-4899-3558-8_31 fatcat:xr5ncxh3knc6bg27z6nspchrmu
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