A Characterization of Finite Ternary Algebras

1997 ◽  
Vol 07 (06) ◽  
pp. 713-721 ◽  
Author(s):  
J. A. Brzozowski ◽  
J. J. Lou ◽  
R. Negulescu
Keyword(s):  
Distributive Lattice ◽  
De Morgan Algebra ◽  
Ternary Algebra ◽  
Set Operations ◽  
Additional Constant ◽  
Ternary Algebras

A ternary algebra is a De Morgan algebra (that is, a distributive lattice with 0 and 1 and a complement operation that satisfies De Morgan's laws) with an additional constant Φ satisfying [Formula: see text], [Formula: see text], and [Formula: see text]. We provide a characterization of finite ternary algebras in terms of "subset-pair algebras," whose elements are pairs (X, Y) of subsets of a given base set ℰ, which have the property X ∪ Y = ℰ, and whose operations are based on common set operations.

FREE TERNARY ALGEBRAS

2000 ◽  
Vol 10 (06) ◽  
pp. 739-749 ◽  
Author(s):  
RAYMOND BALBES
Keyword(s):  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Free Generator ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Bounded Distributive Lattice ◽  
Ternary Algebra ◽  
Free Generators ◽  
Ternary Algebras ◽  
Necessary And Sufficient

A ternary algebra is a bounded distributive lattice with additonal operations e and ~ that satisfies (a+b)~=a~b~, a~~=a, e≤a+a~, e~= e and 0~=1. This article characterizes free ternary algebras by giving necessary and sufficient conditions on a set X of free generators of a ternary algebra L, so that X freely generates L. With this characterization, the free ternary algebra on one free generator is displayed. The poset of join irreducibles of finitely generated free ternary algebras is characterized. The uniqueness of the set of free generators and their pseudocomplements is also established.

INVOLUTED SEMILATTICES AND UNCERTAINTY IN TERNARY ALGEBRAS

2004 ◽  
Vol 14 (03) ◽  
pp. 295-310 ◽  
Author(s):  
J. A. BRZOZOWSKI
Keyword(s):  
Binary Operation ◽  
De Morgan Algebra ◽  
Ternary Algebra ◽  
Ternary Algebras ◽  
Ordered Pairs

An involuted semilattice <S,∨,-> is a semilattice <S,∨> with an involution-: S→S, i.e., <S,∨,-> satisfies [Formula: see text], and [Formula: see text]. In this paper we study the properties of such semilattices. In particular, we characterize free involuted semilattices in terms of ordered pairs of subsets of a set. An involuted semilattice <S,∨,-,1> with greatest element 1 is said to be complemented if it satisfies a∨ā=1. We also characterize free complemented semilattices. We next show that complemented semilattices are related to ternary algebras. A ternary algebra <T,+,*,-,0,ϕ,1> is a de Morgan algebra with a third constant ϕ satisfying [Formula: see text], and (a+ā)+ϕ=a+ā. If we define a third binary operation ∨ on T as a∨b=a*b+(a+b)*ϕ, then <T,∨,-,ϕ> is a complemented semilattice.

A CHARACTERIZATION OF de MORGAN ALGEBRAS

2001 ◽  
Vol 11 (05) ◽  
pp. 525-527 ◽  
Author(s):  
J. A. BRZOZOWSKI
Keyword(s):  
De Morgan Algebra ◽  
De Morgan Algebras ◽  
Ternary Algebras

In this note we show that every de Morgan algebra is isomorphic to a two-subset algebra, < P,⊔,⊓,~,0P,1P>, where P is a set of pairs (X,Y) of subsets of a set I, (X,Y)⊔ (X′,Y′)=(X∩ X′,Y∪ Y′),(X,Y) ⊓ (X′,Y′)=(X∪ X′,Y∩Y′),~(X,Y)= (Y,X), 0P=(I,∅) and 1P=(∅,I). This characterization generalizes a previous result that applied only to a special type of de Morgan algebras called ternary algebras.

On the Partially Ordered Set of Prime Ideals of a Distributive Lattice

1971 ◽  
Vol 23 (5) ◽  
pp. 866-874 ◽  
Author(s):  
Raymond Balbes
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Complete Characterization ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Partially Ordered ◽  
Irreducible Elements

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.

scholarly journals The ideal lattice of an MS-algebra

1988 ◽  
Vol 30 (2) ◽  
pp. 137-143 ◽  
Author(s):  
T. S. Blyth ◽  
J. C. Varlet
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Unary Operation ◽  
Stone Algebra ◽  
Ideal Lattice ◽  
Prime Filter ◽  
De Morgan Algebra ◽  
De Morgan Algebras ◽  
The Ideal

Recently we introduced the notion of an MS-algebra as a common abstraction of a de Morgan algebra and a Stone algebra [2]. Precisely, an MS-algebra is an algebra 〈L; ∧, ∨ ∘, 0, 1〉 of type 〈2, 2, 1, 0, 0〉 such that 〈L; ∧, ∨, 0, 1〉 is a distributive lattice with least element 0 and greatest element 1, and x → x∘ is a unary operation such that x ≤ x∘∘, (x ∧ y)∘ = x∘ ∨ y∘ and 1∘ = 0. It follows that ∘ is a dual endomorphism of L and that L∘∘ = {x∘∘ x ∊ L} is a subalgebra of L that is called the skeleton of L and that belongs to M, the class of de Morgan algebras. Clearly, theclass MS of MS-algebras is equational. All the subvarieties of MS were described in [3]. The lattice Λ (MS) of subvarieties of MS has 20 elements (see Fig. 1) and its non-trivial part (we exclude T, the class of one-element algebras) splits into the prime filter generated by M, that is [M, M1], the prime ideal generated by S, that is [B, S], and the interval [K, K2 ∨ K3].

A NOTE ON HILBERT TERNARY ALGEBRAS

2009 ◽  
Vol 02 (03) ◽  
pp. 367-375 ◽  
Author(s):  
Fatemeh Bahmani
Keyword(s):  
Hilbert Spaces ◽  
Inner Product ◽  
Ternary Algebra ◽  
Positive Multiple ◽  
Ternary Algebras ◽  
Real Complex

We show that every simple abelian real Hilbert ternary algebra is isomorphic to the algebra of Hilbert-Schmidt operators between two real, complex or quaternionic Hilbert spaces, up to a positive multiple of the inner product.

scholarly journals Logics of Involutive Stone Algebras

2021 ◽  
Author(s):  
Sérgio Marcelino ◽  
Umberto Rivieccio
Keyword(s):  
Distributive Lattice ◽  
Independent Interest ◽  
Stone Algebra ◽  
De Morgan Algebra ◽  
De Morgan Algebras ◽  
Finite Set ◽  
Wide Family ◽  
Base Logic

Abstract An involutive Stone algebra (IS-algebra) is a structure that is simultaneously a De Morgan algebra and a Stone algebra (i.e. a pseudo-complemented distributive lattice satisfying the well-known Stone identity, ∼ x ∨ ∼ ∼ x ≈ 1). IS-algebras have been studied algebraically and topologically since the 1980’s, but a corresponding logic (here denoted IS ≤ ) has been introduced only very recently. The logic IS ≤ is the departing point for the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that IS ≤ is a conservative expansion of the Belnap-Dunn four-valued logic (i.e. the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic so as to obtain an extension of IS ≤ . We show that every logic thus defined can be axiomatized by adding a fixed finite set of rule schemata to the corresponding super-Belnap base logic. We also consider a few sample extensions of IS ≤ that cannot be obtained in the above- described way, but can nevertheless be axiomatized finitely by other methods. Most of our axiomatization results are obtained in two steps: through a multiple-conclusion calculus first, which we then reduce to a traditional one. The multiple-conclusion axiomatizations introduced in this process, being analytic, are of independent interest from a proof-theoretic standpoint. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of IS ≤ . Indeed, as in the super-Belnap case, we establish that the finitary extensions of IS ≤ are already uncountably many.

scholarly journals De Morgan Functions and Free De Morgan Algebras

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
Yu. M. Movsisyan ◽  
V. A. Aslanyan ◽  
Alex Manoogian
Keyword(s):  
Boolean Algebra ◽  
Distributive Lattice ◽  
Boolean Functions ◽  
Bounded Lattice ◽  
De Morgan Algebra ◽  
Bounded Distributive Lattice ◽  
Monotone Boolean Functions ◽  
Free Generators ◽  
De Morgan Algebras

AbstractIt is commonly known that the free Boolean algebra on n free generators is isomorphic to the Boolean algebra of Boolean functions of n variables. The free bounded distributive lattice on n free generators is isomorphic to the bounded lattice of monotone Boolean functions of n variables. In this paper, we introduce the concept of De Morgan function and prove that the free De Morgan algebra on n free generators is isomorphic to the De Morgan algebra of De Morgan functions of n variables. This is a solution of the problem suggested by B. I. Plotkin.

PRIME IDEAL CHARACTERIZATION OF STONE ADLs

2010 ◽  
Vol 03 (02) ◽  
pp. 357-367 ◽  
Author(s):  
U. M. Swamy ◽  
S. Ramesh ◽  
Ch. Shanthi Sundar Raj
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper we obtain certain necessary and sufficient conditions for an almost distributive lattice to become a Stone almost distributive lattice in topological and algebraic terms.

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