Pith. Sign in

RECORD

Adjoint operations in twist-products of lattices

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
Ivan Chajda, Helmut Länger

Rings and Algebras · Submitted 3 Jan 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Finally, we introduce another pair \odot and \Rightarrow of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law and we investigate when P_a(L) is closed under these new operations \odot and \Rightarrow.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID2101.00677 (version 1)
Pith ID5ETK454S, Pith's permanent record of this paper and its review
CitationHelmut L\"anger, Ivan Chajda. "Adjoint operations in twist-products of lattices." arXiv:2101.00677 (2021). https://pith.science/paper/5ETK454S
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,473 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.29 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Given an integral commutative residuated lattice L=(L,\vee,\wedge), its full twist-product (L^2,\sqcup,\sqcap) can be endowed with two binary operations \odot and \Rightarrow introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a in L we define a certain subset P_a(L) of L^2. We characterize when P_a(L) is a sublattice of the full twist-product (L^2,\sqcup,\sqcap). In this case P_a(L) together with some natural antitone involution ' becomes a pseudo-Kleene lattice. If L is distributive then (P_a(L),\sqcup,\sqcap,') becomes a Kleene lattice. We present sufficient conditions for P_a(L) being a subalgebra of (L^2,\sqcup,\sqcap,\odot,\Rightarrow) and thus for \odot and \Rightarrow being a pair of adjoint operations on P_a(L). Finally, we introduce another pair \odot and \Rightarrow of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law and we investigate when P_a(L) is closed under these new operations \odot and \Rightarrow.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Given an integral commutative residuated lattice L=(L,\vee,\wedge), its full twist-product (L^2,\sqcup,\sqcap) can be endowed with two binary operations \odot and \Rightarrow introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. background 0.99 0.09 0.04 asserted 0.98
s1 Wille such that it becomes a commutative residuated lattice. background 0.89 0.11 0.05 asserted 0.90
s2 For every a in L we define a certain subset P_a(L) of L^2. contribution 0.76 0.94 0.03 asserted 0.98
s3 We characterize when P_a(L) is a sublattice of the full twist-product (L^2,\sqcup,\sqcap). result 0.54 0.91 0.10 asserted 0.92
s4 In this case P_a(L) together with some natural antitone involution ' becomes a pseudo-Kleene lattice. result 0.99 0.84 0.14 asserted 1.00
s5 If L is distributive then (P_a(L),\sqcup,\sqcap,') becomes a Kleene lattice. result 1.00 0.82 0.12 asserted 1.00
s6 We present sufficient conditions for P_a(L) being a subalgebra of (L^2,\sqcup,\sqcap,\odot,\Rightarrow) and thus for \odot and \Rightarrow being a pair of adjoint operations on P_a(L). contribution 0.69 0.93 0.27 asserted 0.98
s7 Finally, we introduce another pair \odot and \Rightarrow of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law and we investigate when P_a(L) is closed under these new operations \odot and \Rightarrow. contribution 1.00 0.95 0.31 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Given an integral commutative residuated lattice L=(L,\vee,\wedge), its full twist-product (L^2,\sqcup,\sqcap) can be endowed with two binary operations \odot and \Rightarrow introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a in L we define a certain subset P_a(L) of L^2. We characterize when P_a(L) is a sublattice of the full twist-product (L^2,\sqcup,\sqcap). In this case P_a(L) together with some natural antitone involution ' becomes a pseudo-Kleene lattice. If L is distributive then (P_a(L),\sqcup,\sqcap,') becomes a Kleene lattice. We present sufficient conditions for P_a(L) being a subalgebra of (L^2,\sqcup,\sqcap,\odot,\Rightarrow) and thus for \odot and \Rightarrow being a pair of adjoint operations on P_a(L). Finally, we introduce another pair \odot and \Rightarrow of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law and we investigate when P_a(L) is closed under these new operations \odot and \Rightarrow.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 1 sentence without a confident label

Builds on 1 paper
  • The following construction was introduced for distributive lattices in [ref] and generalized for posets by the authors in [ref] : Let $a∈L$ and consider the following subset of $L^2$: [equation] Since our paper [ref] is devoted to posets and not to lattices, we are going to show that if $(P_a(𝐋),⊔,⊓)$ is a sublattice of $(L^2,⊔,⊓)$ then $(P_a(𝐋),⊔,⊓,')$ where the unary operation $'$ on $P_a(𝐋)$ is defined by $(x,y)':=(y,x)$ for all $(x,y)∈P_a(𝐋)$ is a pseudo-Kleene lattice.

    Kleene posets and pseudo-Kleene posets · A construction of pseudo-Kleene lattices in the full twist-product · 0.96

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.