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On Pocrims and Hoops [article]

Rob Arthan, Paulo Oliva
2014 arXiv   pre-print
Pocrims and suitable specialisations thereof are structures that provide the natural algebraic semantics for a minimal affine logic and its extensions.  ...  Hoops comprise a special class of pocrims that provide algebraic semantics for what we view as an intuitionistic analogue of the classical multi-valued Łukasiewicz logic.  ...  ; George Metcalfe for encouraging remarks and for pointers to the literature; and Isabel Ferreirim for helpful correspondence about the theory of hoops.  ... 
arXiv:1404.0816v2 fatcat:2ptuwx3mbjegtlal2usliedixy

Double Negation Semantics for Generalisations of Heyting Algebras

Rob Arthan, Paulo Oliva
2020 Studia Logica: An International Journal for Symbolic Logic  
We consider how each of the above-mentioned translation schemes behaves on two generalisations of Heyting algebras: bounded pocrims and bounded hoops.  ...  We view these as variant semantics and present a semantic formulation of Troelstra's syntactic criteria for a satisfactory negative translation.  ...  : bounded pocrims and bounded hoops.  ... 
doi:10.1007/s11225-020-09909-y fatcat:rdtujdrwdbfofjojk4zx7azimm

(Dual) Hoops Have Unique Halving [chapter]

Rob Arthan, Paulo Oliva
2013 Lecture Notes in Computer Science  
The semantics of these logics can be given using specialisations of algebraic structures known as hoops and coops.  ...  Continuous logic extends the multi-valued Lukasiewicz logic by adding a halving operator on propositions.  ...  for including our problem set in the TPTP Problem Library and for running the problems on a selection of provers; and to Bob Veroff for helping us understand Prover9 performance.  ... 
doi:10.1007/978-3-642-36675-8_9 fatcat:6mg6y3njmfcq5m62yhacg4avmq

(Dual) Hoops Have Unique Halving [article]

Rob Arthan, Paulo Oliva
2013 arXiv   pre-print
Continuous logic extends the multi-valued Lukasiewicz logic by adding a halving operator on propositions.  ...  The semantics of these logics can be given using specialisations of algebraic structures known as hoops.  ...  for including our problem set in the TPTP Problem Library and for running the problems on a selection of provers; and to Bob Veroff for helping us understand Prover9 performance.  ... 
arXiv:1203.0436v2 fatcat:u254uohbbndebo2dhgxo5ozr6q

Hoops, Coops and the Algebraic Semantics of Continuous Logic [article]

Rob Arthan, Paulo Oliva
2012 arXiv   pre-print
Büchi and Owen studied algebraic structures called hoops.  ...  In this paper, we define the notion of continuous hoop, or coop for short, and show that coops provide a natural algebraic semantics for continuous logic.  ...  See [3] for more information on hoops. We say a pocrim is idempotent if it is idempotent as a monoid, i.e., it satisfies x + x = x.  ... 
arXiv:1212.2887v1 fatcat:xnqqxhahb5c2ti6ymp2ilqdh7e

Varieties of Commutative Residuated Integral Pomonoids and Their Residuation Subreducts

Willem J. Blok, James G. Raftery
1997 Journal of Algebra  
Since : and J is an ideal of F, we also have 0r s J. Let s ⌰ . J Ž . We know that 0r s 0r s J, so we have s . Now x, y g so J J n . y . Thus, s l A , hence Ar g S Br , where Br g V V . J J J  ...  If we define [ on B by a [ b s 1 y 1 y a y ḃ˙Ž . ² : Ž a ,bgB then B; [ , y, 0 is a pocrim and, in fact, a hoop.  ...  Ž .x Ž . iii M M * ␤ x, y & ␥ x, y l x, y . n The preceding construction and results extend earlier work on Brouwerian semilattices and hoops.  ... 
doi:10.1006/jabr.1996.6834 fatcat:j7d3lqqifjbjvdtvt7tsyc5iuy

Negative Translations for Affine and Lukasiewicz Logic [article]

Rob Arthan, Paulo Oliva
2019 arXiv   pre-print
On the other hand, the schemes of Glivenko and Gentzen both fail for affine logic, but for different reasons: one can extend affine logic to make Glivenko work and Gentzen fail and vice versa.  ...  We find that in affine logic the translation schemes due to Kolmogorov and G\"odel both satisfy Troelstra's criteria for a negative translation.  ...  We would also like to thank George Metcalfe and Isabel Ferreirim for helpful correspondence.  ... 
arXiv:1912.00012v1 fatcat:cwt5i3lvsndrbjxztk3nry3d3q

Studying Algebraic Structures using Prover9 and Mace4 [article]

Rob Arthan, Paulo Oliva
2019 arXiv   pre-print
The specific tools in our case study are Prover9 and Mace4; the algebraic structures are generalisations of Heyting algebras known as hoops.  ...  We will see that these tools, when combined with human insight and traditional algebraic methods, help us to explore the problem space quickly and effectively.  ...  One might conjecture that any pocrim is a hoop.  ... 
arXiv:1908.06479v1 fatcat:rffhjsvmgbczrnsly5solzqgla

Hoops and their implicational reducts (abstract)

W. Bloki, I. Ferreirim
1993 Banach Center Publications  
A hoop is a pocrim that is naturally ordered; we will denote the class of hoops by HO.  ...  The functor G → P(G), f → f | P (G) establishes an equivalence between the category of Abelian l-groups and l-homomorphisms on the one hand and the category of cancellative hoops and homomorphisms on the  ... 
doi:10.4064/-28-1-219-230 fatcat:kmbco4ykyfhopfvye7bumgqz4q

Page 4414 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
(P-LISBS; Lisbon) On the structure of hoops. (English summary) Algebra Universalis 43 (2000), no. 2-3, 233-257.  ...  The paper is devoted to a study of partially ordered commutative residuated integral monoids (pocrim).  ... 

Free Łukasiewicz and Hoop Residuation Algebras

Joel Berman, W. J. Blok
2004 Studia Logica: An International Journal for Symbolic Logic  
Hoop residuation algebras are the {→, 1}-subreducts of hoops; they include Hilbert algebras and the {→, 1}-reducts of MV-algebras (also known as Wajsberg algebras).  ...  The paper investigates the structure and cardinality of finitely generated algebras in varieties of kpotent hoop residuation algebras.  ...  We refer to [6] for a study of pocrims and for further references. A hoop is a pocrim satisfying in addition x · (x → y) ≈ y · (y → x).  ... 
doi:10.1023/b:stud.0000037125.49866.50 fatcat:7uwcorjqp5ea3kunojqtu7sn6e

A Short Note on Hoops and Continuous {t}-norms

Isabel M. A. Ferreirim
2000 Reports on Mathematical Logic  
It is meant as a modest contribution to the special issue of Reports on Mathematical Logic dedicated to the Workshop on Algebra & Substructural Logic held at the Japan Advanced Institute of Science and  ...  The most relevant references for this note are a paper by Agliano, Ferreirim and Montagna [3] and a recent monograph by Hájek [15] .  ...  A thorough algebraic study of the class of hoops (a.k.a. naturally ordered pocrims) may be found in [5] .  ... 
dblp:journals/rml/Ferreirim00 fatcat:245ds23wu5bb5bh2hjwruuazh4

ŁUKASIEWICZ RESIDUATION ALGEBRAS WITH INFIMUM

A. V. Figallo Jr., A. Figallo, M. Figallo, A. Ziliani
2007 Demonstratio Mathematica  
More precisely, they are algebras (A,-•, 1) of type (2, 2,0) that satisfy: An important subclass of the variety of hoops is the variety of Wajsberg hoops, so named and studied by W. Blok and I.  ...  Büchi and T. Owens ([8]) devoted to a study of Bosbach's algebras, written in the mid-seventies, the commutative members of this equational class were given the name hoops.  ...  Berman and W. Blok ([2] ) investigated the {-1}subreducts of hoops which they called hoop residuation algebras.  ... 
doi:10.1515/dema-2007-0402 fatcat:25nlevpqhzcynelz7rxvaccpn4

An expansion of Basic Logic with fixed points

Luca Spada
2016 Soft Computing - A Fusion of Foundations, Methodologies and Applications  
Moreover, hoops are precisely the partially ordered commutative integral residuated monoids (pocrims) in which the meet operation ∧ is definable by x ∧ y = x · (x ⇒ y).  ...  A comparison between the algebra introduced in Example 2 and the one on next example, can be instructive at this point.  ... 
doi:10.1007/s00500-016-2344-2 fatcat:qisvkcnfcfdcbjefmn5tamutba

Projectivity in (bounded) integral residuated lattices [article]

Paolo Aglianò, Sara Ugolini
2022 arXiv   pre-print
In particular, we obtain results on (Stonean) Heyting algebras, certain varieties of hoops, and product algebras.  ...  Its interaction with divisibility makes our results have a better scope in varieties of divisibile commutative integral residuated lattices, and it allows us to show that many such varieties have the property  ...  In particular, the variety of hoops is a variety of pocrims, and for hoops we have the following result. u → ab = (u → a)(u → b). (1) The two properties above, albeit similar, are quite different in nature  ... 
arXiv:2008.13181v4 fatcat:6z763xftljhsbnhbplmhwuesru
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