Characterizations of the 0-distributive Semilattice
ثبت نشده
چکیده
The 0-distributive semilattice is characterized in terms of semiideals, ideals and filters. Some sufficient conditions and some necessary conditions for 0-distributivity are obtained. Counterexamples are given to prove that certain conditions are not necessary and certain conditions are not sufficient.
منابع مشابه
Modular and Distributive Semilattices
A modular semilattice is a semilattice S in which w > a A ft implies that there exist i,jeS such that x > a. y > b and x A y = x A w. This is equivalent to modularity in a lattice and in the semilattice of ideals of the semilattice, and the condition implies the Kurosh-Ore replacement property for irreducible elements in a semilattice. The main results provide extensions of the classical charac...
متن کاملDistributive Semilattices as Retracts of Ultraboolean Ones; Functorial Inverses without Adjunction
A 〈∨, 0〉-semilattice is ultraboolean, if it is a directed union of finite Boolean 〈∨, 0〉-semilattices. We prove that every distributive 〈∨, 0〉-semilattice is a retract of some ultraboolean 〈∨, 0〉-semilattice. This is established by proving that every finite distributive 〈∨, 0〉-semilattice is a retract of some finite Boolean 〈∨, 0〉-semilattice, and this in a functorial way. This result is, in tu...
متن کاملOn Sturdy Frame of Abstract Algebras
We introduce the notion of a sturdy frame of abstract algebras which is a common generalization of a sturdy semilattice of semigroups, the sum of lattice ordered systems, the strong distributive lattice of semirings, the sturdy frame of type (2, 2) algebras and the strong b-lattice of semirings. Also, we give some properties and characterizations of the sturdy frame of abstract algebras. As an ...
متن کامل. R A ] 4 J an 2 00 6 POSET REPRESENTATIONS OF DISTRIBUTIVE SEMILATTICES
We prove that for any distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (i) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (ii) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) = a ∨ b, then there are a positi...
متن کاملN ov 2 00 7 POSET REPRESENTATIONS OF DISTRIBUTIVE
We prove that for every distributive ∨, 0-semilattice S, there are a meet-semilattice P with zero and a map µ : P × P → S such that µ(x, z) ≤ µ(x, y) ∨ µ(y, z) and x ≤ y implies that µ(x, y) = 0, for all x, y, z ∈ P , together with the following conditions: (P1) µ(v, u) = 0 implies that u = v, for all u ≤ v in P. (P2) For all u ≤ v in P and all a, b ∈ S, if µ(v, u) ≤ a ∨ b, then there are a pos...
متن کامل