Domain Semirings United

Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an endofunction on a semiring that induces a distributive lattice bounded by the two units as its image. This note presents classes of semirings where these approaches coincide.

Extended Green’s relations on Mn(D), a characterization

In this paper, we consider the semiring [Formula: see text] of all [Formula: see text] matrices over a distributive lattice [Formula: see text] and extended Green’s relations [Formula: see text] and [Formula: see text] using [Formula: see text]-ideals. A (left, right) ideal [Formula: see text] of a semiring [Formula: see text] is called a (left, right) [Formula: see text]-ideal if [Formula: see text], where [Formula: see text]. We define [Formula: see text] and [Formula: see text] on a [Formula: see text]-regular semiring [Formula: see text], in which [Formula: see text] is a semilattice, as follows: [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the left [Formula: see text]-ideal generated by [Formula: see text] and [Formula: see text] is the right [Formula: see text]-ideal generated by [Formula: see text]. Here we characterize [Formula: see text] and [Formula: see text] in [Formula: see text] in terms of rows and columns of the matrices.

Logics of Involutive Stone Algebras

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.

Cevian properties in ideal lattices of Abelian ℓ-groups

We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

Weak LI-ideals in implicative almost distributive lattices

In the field of many valued logic, lattice valued logic (especially ideals) plays an important role. Nowadays, lattice valued logic is becoming a research area. Researchers introduced weak LI-ideals of lattice implication algebra. Furthermore, other scholars researched LI-ideals of implicative almost distributive lattice. Therefore, the target of this paper was to investigate new development on the extension of LI-ideal theories and properties in implicative almost distributive lattice. So, in this paper, the notion of weak LI-ideals and maximal weak LI- ideals of implicative almost distributive lattice are defined. The properties of weak LI- ideals in implicative almost distributive lattice are studied and several characterizations of weak LI-ideals are given. Relationship between weak LI-ideals and weak filters are explored. Hence, the extension properties of weak LI-ideal of lattice implication algebra to that of weak LI-ideal of implicative almost distributive lattice were shown.

Subdirect products of an idempotent semiring and a b-lattice of skew-rings

Bandelt and Petrich [Subdirect products of rings and distributive lattices, Proc. Edinburgh Math. Soc. (2) 25(2) (1982) 155–171] characterized a class of additive inverse semirings which are subdirect products of a distributive lattice and a ring. The aim of this paper is to characterize a class of additively regular semirings which are subdirect products of an idempotent semiring and a [Formula: see text]-lattice of skew-rings.

On $n$-Absorbing Ideals in a Lattice

Let L be a lattice, and let n be a positive integer. In this article, we introduce n-absorbing ideals in L. We give some properties of such ideals. We show that every n-absorbing ideal I of L has at most n minimal prime ideals. Also, we give some properties of 2-absorbing and weakly 2-absorbing ideals in L. In particular we show that in every non-zero distributive lattice L, 2-absorbing and weakly 2-absorbing ideals are equivalent.

Smooth digraphs modulo primitive positive constructability and cyclic loop conditions

Finite smooth digraphs, that is, finite directed graphs without sources and sinks, can be partially ordered via pp-constructability. We give a complete description of this poset and, in particular, we prove that it is a distributive lattice. Moreover, we show that in order to separate two smooth digraphs in our poset it suffices to show that the polymorphism clone of one of the digraphs satisfies a prime cyclic loop condition that is not satisfied by the polymorphism clone of the other. Furthermore, we prove that the poset of cyclic loop conditions ordered by their strength for clones is a distributive lattice, too.

L -Fuzzy Prime Spectrums of ADLs

The notion of an Almost Distributive Lattice (ADL) is a common abstraction of several lattice theoretic and ring theoretic generalizations of Boolean algebra and Boolean rings. In this paper, the set of all L -fuzzy prime ideals of an ADL with truth values in a complete lattice L satisfying the infinite meet distributive law is topologized and the resulting space is discussed.

Stability and Median Rationalizability for Aggregate Matchings

We develop the theory of stability for aggregate matchings used in empirical studies and establish fundamental properties of stable matchings including the result that the set of stable matchings is a non-empty, complete, and distributive lattice. Aggregate matchings are relevant as matching data in revealed preference theory. We present a result on rationalizing a matching data as the median stable matching.