Order-theoretic Properties Research Articles

$M_K(T)$-Spaces: Some Order-Theoretical Properties and Applications to Distributive Logics

We present some results involving several ordered structures related to the construction of $M_K(T)$-spaces. These are special closure spaces defined on the basis of a previously given one ${K}$ and were developed by Fern\'andez and Brunetta, 2023. Specifically, we show that for every $T \in {K}$, the poset $\mathbb{RE}_{{K}}(T)$ of weak-relative closure spaces is a sublattice of $\mathbb{CSP}(X)$ (the family of all the spaces with support $X$). On the other hand, it will be shown that the poset $\mathbb{M}({K})$ of all the $M_{{K}}(T)$-spaces does not verify this property, but it is a complete lattice itself. Also, we show in which way some order-theoretic properties are related to the recovery of closure spaces. Finally, we show some applications of $M_{{K}}(T)$-spaces to the class of distributive logics.

LOCC convertibility of entangled states in infinite-dimensional systems

We advance on the conversion of bipartite quantum states via local operations and classical communication (LOCC) for infinite-dimensional systems. We introduce δ-LOCC convertibility based on the observation that any pure state can be approximated by a state with finite-support Schmidt coefficients. We show that δ-LOCC convertibility of bipartite states is fully characterized by a majorization relation between the sequences of squared Schmidt coefficients, providing a novel extension of Nielsen’s theorem for infinite-dimensional systems. Hence, our definition is equivalent to the one of ε-LOCC convertibility (Owari et al 2008 Quantum Inf. Comput. 8 0030), but deals with states having finitely supported sequences of Schmidt coefficients. Additionally, we discuss the notions of optimal common resource and optimal common product in this scenario. The optimal common product always exists, whereas the optimal common resource depends on the existence of a common resource. This highlights a distinction between the resource-theoretic aspects of finite versus infinite-dimensional systems. Our results rely on the order-theoretic properties of majorization for infinite sequences, applicable beyond the LOCC convertibility problem.

Topology of closure systems in algebraic lattices

Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system langle X rangle _P. The closure system langle Y rangle _P generated by the patch closure Y of X is the patch closure of langle X rangle _P. If X is contained in the set of nontrivial prime elements of P then langle X rangle _P is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed.

Properties of congruences of twisted partition monoids and their lattices

We build on the recent characterisation of congruences on the infinite twisted partition monoids P n Φ $\mathcal {P}_{n}^\Phi$ and their finite d $d$ -twisted homomorphic images P n , d Φ $\mathcal {P}_{n,d}^\Phi$ , and investigate their algebraic and order-theoretic properties. We prove that each congruence of P n Φ $\mathcal {P}_{n}^\Phi$ is (finitely) generated by at most ⌈ 5 n 2 ⌉ $\lceil \frac{5n}{2}\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice Cong ( P n Φ ) $\text{\sf Cong}(\mathcal {P}_{n}^\Phi )$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite anti-chains. By way of contrast, the lattice Cong ( P n , d Φ ) $\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )$ is modular but still not distributive for d > 0 $d>0$ , while Cong ( P n , 0 Φ ) $\text{\sf Cong}(\mathcal {P}_{n,0}^\Phi )$ is distributive. We also calculate the number of congruences of P n , d Φ $\mathcal {P}_{n,d}^\Phi$ , showing that the array ( | Cong ( P n , d Φ ) | ) n , d ⩾ 0 $(|\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )|)_{n,d\geqslant 0}$ has a rational generating function, and that for a fixed n $n$ or d $d$ , | Cong ( P n , d Φ ) | $|\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )|$ is a polynomial in d $d$ or n ⩾ 4 $n\geqslant 4$ , respectively.

Smooth p-order ideals in non-unital ordered normed spaces

We introduce smooth p -order ideals ( $$1\le p\le \infty$$ ) to initiate the studies of ideals in non-unital ordered normed spaces. We obtain some order-theoretic properties and examples of these ideals. Furthermore, we show that every semi-M-ideal in affine space A(K) is smooth $$\infty$$ -order ideal. Moreover, we derive that every smooth 1-order ideal is an L-summand in order smooth 1-normed space.

Atomicity and Well Quasi-Order for Consecutive Orderings on Words and Permutations

Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are (a) being atomic, i.e., not being decomposable as a union of two downward closed proper subsets or, equivalently, satisfying the joint embedding property; and (b) being well quasi-ordered. The two posets are (1) words over a finite alphabet under the consecutive subword ordering, and (2) finite permutations under the consecutive subpermutation ordering. Underpinning the four results are characterizations of atomicity and well quasi-order for the subpath ordering on paths of a finite directed graph.

Importation lattices

In recent years, many works have appeared that propose order from basic fuzzy logic connectives. However, all of them assume the connectives to possess some kind of monotonicity, which succinctly implies that the underlying set is already endowed with an order. In this work, given a set P≠∅, we define an algebra based on an implicative-type function I without assuming any order-theoretic properties, either on P or I. Terming it the importation algebra, since the law of importation becomes one of the main axioms in this algebra, we show that such algebras can impose an order on the underlying set P to obtain importation posets. In the case P=[0,1], it is well-known that only R-implications obtained from left-continuous t-norms lead us to richer order-theoretic structures. However, we show here that we can obtain new order-theoretic structures on it even from other families of fuzzy implications, in fact, even when I is not a fuzzy implication, and that one can recover the usual order on [0,1] even from fuzzy implications that do not have the classical ordering property. We also give some sufficient conditions under which such importation posets become lattices. Finally, we present some further possible explorations.

Identifying quantum logics by numerical events

AbstractLetSbe a set of states of a physical system and letp(s) be the probability of an occurrence of an event when the system is in the states∈S. The functionpfromSto [0, 1] is called a numerical event, multidimensional probability or, alternatively,S-probability. Given a set of numerical events which has been obtained by measurements and not supposing any knowledge of the logical structure of the events that appear in the physical system, the question arises which kind of logic is inherent to the system under consideration. In particular, does one deal with a classical situation or a quantum one?In this survey several answers are presented. Starting by associating sets of numerical events to quantum logics we study structures that arise whenS-probabilities are partially ordered by the order of functions and characterize those structures which indicate that one deals with a classical system. In particular, sequences of numerical events are considered that give rise to Bell-like inequalities. At the center of all studies there are so called algebras ofS-probabilities, subsets of these and their generalizations. A crucial feature of these structures is that order theoretic properties can be expressed by the addition and subtraction of real functions entailing simplified algorithmic procedures.The study of numerical events and algebras ofS-probabilities goes back to a cooperation of E. G. Beltrametti and M. J. Mączyński in 1991 and has since then resulted in a series of subsequent papers of physical interest the main results of which will be commented on and put in an appropriate context.

The poset of unlabeled induced subgraphs of a finite graph

The set of all unlabeled induced subgraphs of a finite graph G can be made into a poset by defining $$H_1 \le H_2$$ iff $$H_1$$ is a subgraph of $$H_2$$. This paper discusses some of the connections between the graph properties of G and the order theoretic properties of this poset. This paper then restricts this class of subgraphs to only connected unlabeled induced subgraphs and looks at some of the order theoretic properties of that poset.

CM-ideals and L1-matricial split faces

We discuss the order-theoretic properties of CM-ideals in matricially order smooth∞-normed spaces. We study the relation between CM-ideals and CL-summands in the matrix duality setup. We introduce the notion of L1-matricial split faces in an L1-matricially normed space and characterize CM-ideals in a matricially order smooth∞-normed space Vin terms of the L1-matricial split face of the L1-matrix convex set{Qn(V)}.

Algorithmic correspondence and canonicity for non-distributive logics

We extend the theory of unified correspondence to a broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as ‘lattices with operators’. Specifically, we introduce a syntactic definition of the class of Sahlqvist formulas and inequalities which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. We also introduce the algorithm ALBA, parametric in each LE-setting, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. Projecting these results on specific signatures yields state-of-the-art correspondence and canonicity theory for many well known modal expansions of classical and intuitionistic logic and for substructural logics, from classical poly-modal logics to (bi-)intuitionistic modal logics to the Lambek calculus and its extensions, the Lambek-Grishin calculus, orthologic, the logic of (not necessarily distributive) De Morgan lattices, and the multiplicative-additive fragment of linear logic.

Domains of commutative C*-subalgebras

AbstractA C*-algebra is determined to a great extent by the partial order of its commutative C*-subalgebras. We study order-theoretic properties of this directed-complete partially ordered (dcpo). Many properties coincide: the dcpo is, equivalently, algebraic, continuous, meet-continuous, atomistic, quasi-algebraic or quasi-continuous, if and only if the C*-algebra is scattered. For C*-algebras with enough projections, these properties are equivalent to finite-dimensionality. Approximately finite-dimensional elements of the dcpo correspond to Boolean subalgebras of the projections of the C*-algebra. Scattered C*-algebras are finitedimensional if and only if their dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and only if their dcpo is order-scattered.

A STUDY OF TRUTH PREDICATES IN MATRIX SEMANTICS

AbstractAbstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic ${\cal L}$ is associated with a matrix semantics $Mo{d^{\rm{*}}}{\cal L}$. This article is a contribution to the systematic study of the so-called truth sets of the matrices in $Mo{d^{\rm{*}}}{\cal L}$. In particular, we show that the fact that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of ${\cal L}$. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ are implicitly definable if and only if the Leibniz operator is injective on deductive filters of ${\cal L}$ over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of ${\cal L}$ to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in $Mo{d^{\rm{*}}}{\cal L}$ that corresponds to the order-reflection of the Leibniz operator.

Order-theoretic properties and separability of some sets of quasi-measures

Let {{mathfrak {M}}} and {{mathfrak {R}}} be algebras of subsets of a set Omega with {mathfrak {M}}subset {mathfrak {R}}. Denote by E(mu ) the set of all quasi-measure extensions of a given quasi-measure mu on {mathfrak {M}} to {mathfrak {R}}. We present some results on the coincidence of the bands, in {mathfrak {R}}, generated by E(mu ) and extr E(mu ). Moreover, we show that if mu is atomic, then E(mu ) is contained in a principal band in {mathfrak {R}} if and only if it is separable. Another sufficient condition for the separability of E(mu ) is also presented.

Order-theoretic properties of some sets of quasi-measures

Let $\mathfrak M$ and $\mathfrak R$ be algebras of subsets of a set $\Omega $ with $\mathfrak M\subset \mathfrak R$, and denote by $E(\mu )$ the set of all quasi-measure extensions of a given quasi-measure $\mu $ on $\mathfrak M$ to $\mathfrak R$. We show that $E(\mu )$ is order bounded if and only if it is contained in a principal ideal in $ba(\mathfrak R)$ if and only if it is weakly compact and $\operatorname{extr} E(\mu )$ is contained in a principal ideal in $ba(\mathfrak R)$. We also establish some criteria for the coincidence of the ideals, in $ba(\mathfrak R)$, generated by $E(\mu )$ and $\operatorname{extr} E(\mu )$.

Canonicity results for mu-calculi: an algorithmic approach

We investigate the canonicity of inequalities of the intuitionistic mu-calculus. The notion of canonicity in the presence of fixed point operators is not entirely straightforward. In the algebraic setting of canonical extensions we examine both the usual notion of canonicity and what we will call tame canonicity. This latter concept has previously been investigated for the classical mu-calculus by Bezhanishvili and Hodkinson. Our approach is in the spirit of Sahlqvist theory. That is, we identify syntactically-defined classes of inequalities, namely the restricted inductive and tame inductive inequalities, which are, respectively, canonical or tame canonical. Our approach is to use an algorithm which processes inequalities with the aim of eliminating propositional variables. The algorithm we introduce is closely related to the algorithms ALBA and mu-ALBA studied by Conradie |$et\,al$|⁠. It is based on a calculus of rewrite rules, the soundness of which rests upon the way in which algebras embed into their canonical extensions and the order-theoretic properties of the latter. We show that the algorithm succeeds on every restricted inductive inequality by means of a so-called proper run, and that this is sufficient to guarantee their canonicity. Likewise, we are able to show that the algorithm succeeds on every tame inductive inequality by means of a so-called tame run. In turn, this guarantees their tame canonicity.

Projectable ℓ-groups and algebras of logic: Categorical and algebraic connections

P.F. Conrad and other authors launched a general program for the investigation of lattice-ordered groups, aimed at elucidating some order-theoretic properties of these algebras by inquiring into the structure of their lattices of convex ℓ-subgroups. This approach can be naturally extended to residuated lattices and their convex subalgebras. In this broader perspective, we revisit the Galatos–Tsinakis categorical equivalence between integral generalized MV algebras and negative cones of ℓ-groups with a nucleus, showing that it restricts to an equivalence of the full subcategories whose objects are the projectable members of these classes. Upon recalling that projectable integral generalized MV algebras and negative cones of projectable ℓ-groups can be endowed with a positive Gödel implication, and turned into varieties by including this implication in their signature, we prove that there is an adjunction between the categories whose objects are the members of these varieties and whose morphisms are required to preserve implications.

Banach lattice algebras: some questions, but very few answers

We pose a number of questions and problems about Banach lattice algebras. These concern: What should the definition be? How to add an identity. Order theoretic properties of the multiplication. Order theoretic properties of the left regular representation.

Classifying Finite-Dimensional C*-Algebras by Posets of Their Commutative C*-Subalgebras

We consider the functor C that to a unital C*-algebra A assigns the partial order set C(A) of its commutative C*-subalgebras ordered by inclusion. We investigate how some C*-algebraic properties translate under the action of C to order-theoretical properties. In particular, we show that A is finite dimensional if and only C(A) satisfies certain chain conditions. We eventually show that if A and B are C*-algebras such that A is finite dimensional and C(A) and C(B) are order isomorphic, then A and B must be *-isomorphic.

A Parametrization of the Irreducible Representations of a Compact Inverse Semigroup

The aim in this article is to provide a parametrization of the finite dimensional irreducible representations of a compact inverse semigroup in terms of the irreducible representations of maximal subgroups and order theoretic properties of the idempotent set. As a consequence, we obtain a new, and more conceptual, proof of the following theorem of Shneperman: a compact inverse semigroup has enough finite dimensional irreducible representations to separate points if and only if its idempotent set is totally disconnected. Moreover, we also prove that every norm continuous irreducible *-representation of a compact inverse semigroup on a Hilbert space is finite dimensional.