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Prime Filters and Ideals in Distributive Lattices

Adam Grabowski (2013)

Formalized Mathematics

The article continues the formalization of the lattice theory (as structures with two binary operations, not in terms of ordering relations). In the Mizar Mathematical Library, there are some attempts to formalize prime ideals and filters; one series of articles written as decoding [9] proven some results; we tried however to follow [21], [12], and [13]. All three were devoted to the Stone representation theorem [18] for Boolean or Heyting lattices. The main aim of the present article was to bridge...

Prime ideal theorem for double Boolean algebras

Léonard Kwuida (2007)

Discussiones Mathematicae - General Algebra and Applications

Double Boolean algebras are algebras (D,⊓,⊔,⊲,⊳,⊥,⊤) of type (2,2,1,1,0,0). They have been introduced to capture the equational theory of the algebra of protoconcepts. A filter (resp. an ideal) of a double Boolean algebra D is an upper set F (resp. down set I) closed under ⊓ (resp. ⊔). A filter F is called primary if F ≠ ∅ and for all x ∈ D we have x ∈ F or x ⊲ ∈ F . In this note we prove that if F is a filter and I an ideal such that F ∩ I = ∅ then there is a primary filter G containing F such that G...

Prime ideals in 0-distributive posets

Vinayak Joshi, Nilesh Mundlik (2013)

Open Mathematics

In the first section of this paper, we prove an analogue of Stone’s Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization...

Prime ideals in the lattice of additive induced-hereditary graph properties

Amelie J. Berger, Peter Mihók (2003)

Discussiones Mathematicae Graph Theory

An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups,...

Prime, weakly prime and almost prime elements in multiplication lattice modules

Emel Aslankarayigit Ugurlu, Fethi Callialp, Unsal Tekir (2016)

Open Mathematics

In this paper, we study multiplication lattice modules. We establish a new multiplication over elements of a multiplication lattice module.With this multiplication, we characterize idempotent element, prime element, weakly prime element and almost prime element in multiplication lattice modules.

Primeness and semiprimeness in posets

Vilas S. Kharat, Khalid A. Mokbel (2009)

Mathematica Bohemica

The concept of a semiprime ideal in a poset is introduced. Characterizations of semiprime ideals in a poset P as well as characterizations of a semiprime ideal to be prime in P are obtained in terms of meet-irreducible elements of the lattice of ideals of P and in terms of maximality of ideals. Also, prime ideals in a poset are characterized.

Primes, coprimes and multiplicative elements

Melvin F. Janowitz, Robert C. Powers, Thomas Riedel (1999)

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to study conditions under which the restriction of a certain Galois connection on a complete lattice yields an isomorphism from a set of prime elements to a set of coprime elements. An important part of our study involves the set on which the way-below relation is multiplicative.

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