Quasi-decompositions and quasidirect products of Hilbert algebras

2021 ◽  
Vol 71 (4) ◽  
pp. 781-806
Author(s):  
Jānis Cīrulis
Keyword(s):  
Semidirect Product ◽  
Universal Property ◽  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
Product Construction ◽  
Exact Sequences

Abstract A quasi-decomposition of a Hilbert algebra A is a pair (C, D) of its subalgebras such that (i) every element a ∈ A is a meet c ∧ d with c ∈ C, d ∈ D, where c and d are compatible (i.e., c → d = c → (c ∧ d)), and (ii) d → c = c (then c is uniquely defined). Quasi-decompositions are intimately related to the so-called triple construction of Hilbert algebras, which we reinterpret as a construction of quasidirect products. We show that it can be viewed as a generalization of the semidirect product construction, that quasidirect products has a certain universal property and that they can be characterised in terms of short exact sequences. We also discuss four classes of Hilbert algebras and give for each of them conditions on a quasi-decomposition of an arbitrary Hilbert algebra A under which A belongs to this class.

scholarly journals Square-integrable representations of non-unimodular groups

1976 ◽  
Vol 15 (1) ◽  
pp. 1-12 ◽  
Author(s):  
A.L. Carey
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Hilbert Algebra ◽  
Technical Tool ◽  
Integrable Representation ◽  
Hilbert Algebras ◽  
Orthogonality Relations ◽  
Square Integrable Representation ◽  
Main Technical Tool

In the last three years a number of people have investigated the orthogonality relations for square integrable representations of non-unimodular groups, extending the known results for the unimodular case. The results are stated in the language of left (or generalized) Hilbert algebras. This paper is devoted to proving the orthogonality relations without recourse to left Hilbert algebra techniques. Our main technical tool is to realise the square integrable representation in question in a reproducing kernel Hilbert space.

Implicational formulas in intuitionistic logic

1974 ◽  
Vol 39 (4) ◽  
pp. 661-664 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Propositional Calculus ◽  
Modus Ponens ◽  
Equivalence Classes ◽  
Alternative Proof ◽  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
Free Generators ◽  
Finite Set ◽  
Image Position ◽  
The Right

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P. He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra In on n free generators. In the present paper we give an alternative proof of the finiteness of In, and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that In is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1].Let V be a finite set of propositional variables, and let F(V) be the set of all formulas built up from V ⋃ {t} using → alone. The algebra defined on the equivalence classes , by settingis a free Hilbert algebra I(V) on the free generators . A set T ⊆ F(V) is a theory if ⊦pA implies A ∈ T, and T is closed under modus ponens. For T a theory, T[A] is the theory {B ∣ A → B ∈ T}. A theory T is p-prime, where p ∈ V, if p ∉ T and, for any A ∈ F(V), A ∈ T or A → p ∈ T. A theory is prime if it is p-prime for some p. Pp(V) denotes the set of p-prime theories in F(V), P(V) the set of prime theories. T ∈ P(V) is minimal if there is no theory in P(V) strictly contained in T. Where X = {A1, …, An} is a finite set of formulas, let X → B be A1 →····→·An → B (ϕ → B is B). A formula A is a p-formula if p is the right-most variable occurring in A, i.e. if A is of the form X → p.

scholarly journals On GE-Algebras

2020 ◽  
Author(s):  
Ravikumar Bandaru ◽  
Arsham Borumand Saeid ◽  
Young Bae Jun
Keyword(s):  
Intuitionistic Logic ◽  
Algebraic Structure ◽  
Classical Logic ◽  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
Exchange Algebra ◽  
Generalized Exchange

Hilbert algebras are important tools for certain investigations in intuitionistic logic and other non-classical logic and as a generalization of Hilbert algebra a new algebraic structure, called a GE-algebra (generalized exchange algebra), is introduced and studied its properties. We consider filters, upper sets and congruence kernels in a GE-algebra. We also characterize congruence kernels of transitive GE-algebras.

Hilbert algebras with supremum generated by finite chains

2019 ◽  
Vol 69 (4) ◽  
pp. 953-963
Author(s):  
Hernando Gaitán
Keyword(s):  
Topological Space ◽  
Dual Space ◽  
Hilbert Algebra ◽  
Hilbert Algebras

Abstract Based on the work of A. Monteiro, A. Torrens, and D. Buşneag, in this paper we point out that the dual space of Hilbert algebras with supremum generated by chains depends, modulo the dual space of a Hilbert algebra with supremum defined by S. Celani an D. Montangie, exclusively, on the order carried out by the topological space. We use such a characterization to prove that a bounded Hilbert algebra generated by chains is determined by the monoid of its endomorphisms.

Fuzzy filters of Sheffer stroke Hilbert algebras

2021 ◽  
Vol 40 (1) ◽  
pp. 759-772
Author(s):  
Tahsin Oner ◽  
Tugce Katican ◽  
Arsham Borumand Saeid
Keyword(s):  
Cartesian Product ◽  
Hilbert Algebra ◽  
Fuzzy Filter ◽  
Hilbert Algebras ◽  
Sheffer Stroke ◽  
Fuzzy Filters

The aim of this study is to introduce fuzzy filters of Sheffer stroke Hilbert algebra. After defining fuzzy filters of Sheffer stroke Hilbert algebra, it is shown that a quotient structure of this algebra is described by its fuzzy filter. In addition to this, the level filter of a Sheffer stroke Hilbert algebra is determined by its fuzzy filter. Some fuzzy filters of a Sheffer stroke Hilbert algebra are defined by a homomorphism. Normal and maximal fuzzy filters of a Sheffer stroke Hilbert algebra and the relation between them are presented. By giving the Cartesian product of fuzzy filters of a Sheffer stroke Hilbert algebra, various properties are examined.

scholarly journals Study of Hilbert algebras in point of filters

2016 ◽  
Vol 24 (2) ◽  
pp. 221-251
Author(s):  
Ali Soleimani Nasab ◽  
Arsham Borumand Saeid
Keyword(s):  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
The Relationship

Abstract The aim of this work is to introduce some types of filters in Hilbert algebras. Some theorems are stated and proved which determine the relationship between these notions and other filters of Hilbert algebra and by some examples we show that these concepts are different. The relationships between these filters and quotient algebras that are constructed via these filters are described.

scholarly journals On Fuzzy Deductive Systems of Hilbert Algebras

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Gezahagne Mulat Addis ◽  
Derso Abeje Engidaw
Keyword(s):  
Fuzzy Set ◽  
Complete Lattice ◽  
Algebraic Closure ◽  
Lattice Isomorphism ◽  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
Deductive Systems ◽  
Congruence Relations ◽  
Fuzzy Congruence ◽  
Distributive Law

In this paper, we study fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law. Several characterizations are obtained for fuzzy deductive systems generated by a fuzzy set. It is also proved that the class of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system. Furthermore, we obtain a lattice isomorphism between the class of fuzzy deductive systems and the class of fuzzy congruence relations in the variety of Hilbert algebras.

scholarly journals Hilbert Algebras of Fractions

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Christina-Theresia Dan
Keyword(s):  
Closed Subset ◽  
Sufficient Condition ◽  
Hilbert Algebra ◽  
Necessary And Sufficient Condition ◽  
Inductive Limits ◽  
Hilbert Algebras ◽  
Deductive Systems ◽  
Necessary And Sufficient

Let be a bounded Hilbert algebra and a -closed subset of . The Hilbert algebra of fractions is studied regarding maximal and irreducible deductive systems. As important results, we can mention a necessary and sufficient condition for a Hilbert algebra of fractions to be local and the characterization of this kind of algebras as inductive limits of some particular directed systems.

scholarly journals A note on homomorphisms of Hilbert algebras

2002 ◽  
Vol 29 (1) ◽  
pp. 55-61 ◽  
Author(s):  
Sergio Celani
Keyword(s):  
Representation Theorem ◽  
Separation Theorem ◽  
Boolean Algebras ◽  
Order Ideal ◽  
Hilbert Algebra ◽  
Ordered Sets ◽  
Hilbert Algebras ◽  
Deductive Systems ◽  
Similar Notion

We give a representation theorem for Hilbert algebras by means of ordered sets and characterize the homomorphisms of Hilbert algebras in terms of applications defined between the sets of all irreducible deductive systems of the associated algebras. For this purpose we introduce the notion of order-ideal in a Hilbert algebra and we prove a separation theorem. We also define the concept of semi-homomorphism as a generalization of the similar notion of Boolean algebras and we study its relation with the homomorphism and with the deductive systems.

Sign in / Sign up

Export Citation Format

Share Document