Finite IUML-algebras, Finite Forests and Orthopairs

Aguzzoli, Stefano; Boffa, Stefania; Ciucci, Davide; Gerla, Brunella

doi:10.3233/fi-2018-1735

Cited by 5 publications

(5 citation statements)

References 16 publications

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“…A clear candidate is the variety of Nilpotent Minimum algebras (NM-algebras for short) that, similarly to Gödel algebras, is dual to a forest-based category (see [6]). Another class of algebras that is dual to forests is that of IUML-algebras, studied in [1]. The latter are algebras based on uninorms rather than t-norms, but similar methods to those developed in the present chapter might be used to define and study modal operators on them.…”

Section: Discussionmentioning

confidence: 96%

“…Let R and R ♦ be as in Section 4, and put R = R ∩ R ♦ . Then, by Proposition 4.3 (1), it follows that (F A , R) is a basic frame.…”

Section: Adding Structure To Gödel Algebras With Operatorsmentioning

confidence: 88%

“…x ∨ y = x ∪ y (the set-theoretic union); 3. x → y = (↑(x \ y)) c = F \ ↑(x \ y), where \ denotes the set-theoretical difference, for every z ∈ F , ↑z = {k ∈ F | k ≥ z} and c denotes the set-theoretical complement 1. …”

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confidence: 99%

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Algebras and relational frames for Gödel modal logic and some of its extensions

Flaminio¹,

Godo²,

Menchón³

et al. 2021

Preprint

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Gödel modal logics can be seen as extenions of intutionistic modal logics with the prelinearity axiom. In this paper we focus on the algebraic and relational semantics for Gödel modal logics that leverages on the duality between finite Gödel algebras and finite forests, i.e. finite posets whose principal downsets are totally ordered. We consider different subvarieties of the basic variety GAO of Gödel algebras with two modal operators (GAOs for short) and their corresponding classes of forest frames, either with one or two accessibility relations. These relational structures can be considered as prelinear versions of the usual relational semantics of intuitionistic modal logic. More precisely we consider two main extensions of finite Gödel algebras with operators: the one obtained by adding Dunn axioms, typically studied in the fragment of positive classical (and intuitionistic) logic, and the one determined by adding Fischer Servi axioms. We present Jónsson-Tarski like representation theorems for the different types of finite GAOs considered in the paper.

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Section: Discussionmentioning

confidence: 96%

“…Let R and R ♦ be as in Section 4, and put R = R ∩ R ♦ . Then, by Proposition 4.3 (1), it follows that (F A , R) is a basic frame.…”

Section: Adding Structure To Gödel Algebras With Operatorsmentioning

confidence: 88%

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Algebras and relational frames for Gödel modal logic and some of its extensions

Flaminio¹,

Godo²,

Menchón³

et al. 2021

Preprint

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“…We prove that (A, B) = (X 1 , X 2 ). It is trivial that A ⊆ X 1 and B ⊆ X 2 . Now, we suppose that N ∈ X 1 , and we intend to prove that 13), and so M ⊆ N .…”

Section: Propositionmentioning

confidence: 99%

“…We exclude the operations ⊙ 4 and ֒→ 4 , since they can not be obtained starting from operations between the orthopairs 2. By 15, i ϕ ∧ i ψ = i (ϕ ∧ ψ) 3.…”

mentioning

confidence: 99%

Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects

Boffa

Gerla

2020

Lecture Notes in Computer Science

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In this thesis, a generalization of the classical Rough set theory [81] is developed considering the so-called sequences of orthopairs that we define in [16] as special sequences of rough sets.Mainly, our aim is to introduce some operations between sequences of orthopairs, and to discover how to generate them starting from the operations concerning standard rough sets (defined in [29]). Also, we prove several representation theorems representing the class of finite centered Kleene algebras with the interpolation property [28], and some classes of finite residuated lattices (more precisely, we consider Nelson algebras [86], Nelson lattices [21], IUML-algebras [69] and Kleene lattice with implication [24]) as sequences of orthopairs. I am deeply thankful to my family: my parents, my sister Rosa, my brother Angelo and his girlfriend Roberta, for their unconditional love and spiritual support in all my life. v vii "I magination is the Discovering Faculty, pre-eminently. It is that which penetrates into the unseen worlds around us, the worlds of Science. It is that which feels & discovers what is, the real which we see not, which exists not for our senses. Those who have learned to walk on the threshold of the unknown worlds, by means of what are commonly termed par excellence the exact sciences, may then with the fair white wings of Imagination hope to soar further into the unexplored amidst which we live.

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