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- Backward-shift operator (1)
- Beurling-Lax theorem (1)
- Boolean Algebras (1)
- Bunched implication algebras (1)
- Cuntz relations (1)
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- De Branges-Rovnyak spaces (1)
- Distributive lattice-ordered magmas (1)
- Enumerating finite models (1)
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- Idempotent semirings (1)
- Injective and projective semimodules (1)
- Involutive residuated lattices (1)
- Local Finiteness (1)
- Pregroups (1)
- Rational functions (1)
- Representations (1)
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- Semirings (1)
- Structure theorems (1)
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- ℓ-pregroups (1)
Articles 1 - 9 of 9
Full-Text Articles in Algebra
Algorithmic Correspondence For Relevance Logics, Bunched Implication Logics, And Relation Algebras Via An Implementation Of The Algorithm Pearl, Willem Conradie, Valentin Goranko, Peter Jipsen
Algorithmic Correspondence For Relevance Logics, Bunched Implication Logics, And Relation Algebras Via An Implementation Of The Algorithm Pearl, Willem Conradie, Valentin Goranko, Peter Jipsen
Mathematics, Physics, and Computer Science Faculty Articles and Research
The non-deterministic algorithmic procedure PEARL (acronym for ‘Propositional variables Elimination Algorithm for Relevance Logic’) has been recently developed for computing first-order equivalents of formulas of the language of relevance logics LR in terms of the standard Routley-Meyer relational semantics. It succeeds on a large class of axioms of relevance logics, including all so called inductive formulas. In the present work we re-interpret PEARL from an algebraic perspective, with its rewrite rules seen as manipulating quasi-inequalities interpreted over Urquhart’s relevant algebras, and report on its recent Python implementation. We also show that all formulae on which PEARL succeeds are canonical, i.e., …
Unary-Determined Distributive ℓ -Magmas And Bunched Implication Algebras, Natanael Alpay, Peter Jipsen, Melissa Sugimoto
Unary-Determined Distributive ℓ -Magmas And Bunched Implication Algebras, Natanael Alpay, Peter Jipsen, Melissa Sugimoto
Mathematics, Physics, and Computer Science Faculty Articles and Research
A distributive lattice-ordered magma (dℓ-magma) (A,∧,∨,⋅) is a distributive lattice with a binary operation ⋅ that preserves joins in both arguments, and when ⋅ is associative then (A,∨,⋅) is an idempotent semiring. A dℓ-magma with a top ⊤ is unary-determined if x⋅y=(x⋅⊤∧y)∨(x∧⊤⋅y). These algebras are term-equivalent to a subvariety of distributive lattices with ⊤ and two join-preserving unary operations p, q. We obtain simple conditions on p, q such that x⋅y=(px∧y)∨(x∧qy) is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the …
Generalized Grassmann Algebras And Applications To Stochastic Processes, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler
Generalized Grassmann Algebras And Applications To Stochastic Processes, Daniel Alpay, Paula Cerejeiras, Uwe Kaehler
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper, we present the groundwork for an Itô/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmetry with Z3-graded algebras. To this end, we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.
The Structure Of Finite Commutative Idempotent Involutive Residuated Lattices, Peter Jipsen, Olim Tuyt, Diego Valota
The Structure Of Finite Commutative Idempotent Involutive Residuated Lattices, Peter Jipsen, Olim Tuyt, Diego Valota
Mathematics, Physics, and Computer Science Faculty Articles and Research
We characterize commutative idempotent involutive residuated lattices as disjoint unions of Boolean algebras arranged over a distributive lattice. We use this description to introduce a new construction, called gluing, that allows us to build new members of this variety from other ones. In particular, all finite members can be constructed in this way from Boolean algebras. Finally, we apply our construction to prove that the fusion reduct of any finite member is a distributive semilattice, and to show that this variety is not locally finite.
Conjunctive Join-Semilattices, Charles N. Delzell, Oghenetega Ighedo, James J. Madden
Conjunctive Join-Semilattices, Charles N. Delzell, Oghenetega Ighedo, James J. Madden
Mathematics, Physics, and Computer Science Faculty Articles and Research
A join-semilattice L with top is said to be conjunctive if every principal ideal is an intersection of maximal ideals. (This is equivalent to a first-order condition in the language of semilattices.) In this paper, we explore the consequences of the conjunctivity hypothesis for L, and we define and study a related property, called “ideal conjunctivity,” which is applicable to join-semilattices without top. Results include the following: (a) Every conjunctive join-semilattice is isomorphic to a join-closed subbase for a compact T1-topology on max L, the set of maximal ideals of L, and under weak hypotheses …
Beurling-Lax Type Theorems And Cuntz Relations, Daniel Alpay, Fabrizio Colombo, Irene Sabadini, Baruch Schneider
Beurling-Lax Type Theorems And Cuntz Relations, Daniel Alpay, Fabrizio Colombo, Irene Sabadini, Baruch Schneider
Mathematics, Physics, and Computer Science Faculty Articles and Research
We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.
Injective And Projective Semimodules Over Involutive Semirings, Peter Jipsen, Sara Vanucci
Injective And Projective Semimodules Over Involutive Semirings, Peter Jipsen, Sara Vanucci
Mathematics, Physics, and Computer Science Faculty Articles and Research
We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval [d,1] to be a subalgebra of an involutive residuated lattice, where d is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for …
Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, I. Sabadini, Daniele C. Struppa
Total Differentiability And Monogenicity For Functions In Algebras Of Order 4, I. Sabadini, Daniele C. Struppa
Mathematics, Physics, and Computer Science Faculty Articles and Research
In this paper we discuss some notions of analyticity in associative algebras with unit. We also recall some basic tool in algebraic analysis and we use them to study the properties of analytic functions in two algebras of dimension four that played a relevant role in some work of the Italian school, but that have never been fully investigated.
Lattice-Ordered Pregroups Are Semi-Distributive, Nikolaos Galatos, Peter Jipsen, Michael Kinyon, Adam Přenosil
Lattice-Ordered Pregroups Are Semi-Distributive, Nikolaos Galatos, Peter Jipsen, Michael Kinyon, Adam Přenosil
Mathematics, Physics, and Computer Science Faculty Articles and Research
We prove that the lattice reduct of every lattice-ordered pregroup is semidistributive. This is a consequence of a certain weak form of the distributive law which holds in lattice-ordered pregroups.