For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P ) = Id(P )−{∅}. This note obtains various results to the effect that Id(P ) is always, and id(P ) often, “essentially larger” than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone...

This paper shows that any compactly generated lattice is a subdirect product of subdirectly irreducible lattices which are complete and upper continuous. An example of a compactly generated lattice which cannot be subdirectly decomposed into subdirectly irreducible compactly generated lattices is given. In the case of an ideal lattice of a lattice L, the decomposition into subdirectly irreducib...

Let $R$ be a commutative Noetherian ring with non-zero identity, $fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we first study the membership of $Ext^{s+t}_{R}(N, X)$ and $Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of the category of $R$--modules. Then, we present some conditions which ensure the exi...

Let P := k [X,Y, Z] be a polynomial ring over an algebraic closed field k and (X, Y , Z, XY Z, XY Z ) · k [X,Y, Z] an (X,Y, Z)-primary ideal in P , (m, n, l, a, b, c, d, e, f are integers). The ideal Q=(X, Y , Z, XY Z, XY Z ) ·R is (X,Y, Z) · R-primary ideal in the local ring R = k [X,Y, Z](x,y,z). In this short note we give a formula for the calculation of Samuel multiplicity e0(Q,R) of the id...

In this paper, we introduce a coupled N -structure which is the generalization of N -structure. Using this coupled N -structure, we have applied in a subtraction algebra and have introduced the notion of a coupled N -subalgebra, a coupled N -ideal. Also the characterization of coupled N -ideal is presented.

In this work, we attempt to investigate the connection between various types of ideals (for examples (m,n)-ideal, bi-ideal, interior-ideal, quasi-ideal, prime-ideal and maximal-ideal) of an ordered semigroup (S, ·,≤) and the corresponding, hyperideals of its EL-hyperstructure (S, ∗) (if exists). Moreover, we construct the class of EL-Γ-semihypergroup, associated to a partially-ordered Γsemigroup.

We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. The...

Let $R$ be a commutative ring and $M$ be an $R$-module. In this paper, we investigate some properties of 2-absorbing submodules of $M$. It is shown that $N$ is a 2-absorbing submodule of $M$ if and only if whenever $IJLsubseteq N$ for some ideals $I,J$ of R and a submodule $L$ of $M$, then $ILsubseteq N$ or $JLsubseteq N$ or $IJsubseteq N:_RM$. Also, if $N$ is a 2-absorbing submodule of ...

Given an ideal I on ω let a(I) (ā(I)) be minimum of the cardinalities of in nite (uncountable) maximal I-almost disjoint subsets of [ω]. We show that a(Ih) > ω if Ih is a summable ideal; but a(Z~ μ) = ω for any tall density ideal Z~ μ including the density zero ideal Z. On the other hand, you have b ≤ ā(I) for any analytic P -ideal I, and ā(Z~ μ) ≤ a for each density ideal Z~ μ. For each ideal ...