Non-trivial Equivalence Relation Research Articles

Subalgebra lattices of totally reflexive sub-preprimal algebras

For a set Q of relations on a finite set A, the set $$\mathrm{Pol}_{A}Q$$ , of all operations on A preserving all relations in Q, is a clone. The set of all clones on a given set forms a lattice under inclusion. Each of its maximal elements can be represented as $$\mathrm{Pol}_{A}\{\rho \}$$ where $$\rho $$ is one of the six types of relations determined by Ivo G. Rosenberg. Four types of them are totally reflexive. These relations are bounded orders, non-trivial equivalence relations, central relations, and universal relations. An algebra $$\underline{A}$$ is said to be totally reflexive sub-preprimal if its clone of term operations is where $$\rho _{1}$$ and $$\rho _{2}$$ are totally reflexive relations. We describe the subalgebra lattices of such algebras.

Reasoning about graded strategy quantifiers

In this paper we introduce and study Graded Strategy Logic (GSL), an extension of Strategy Logic (SL) with graded quantifiers. SL is a powerful formalism that allows to describe useful game concepts in multi-agent settings by explicitly quantifying over strategies treated as first-order citizens. In GSL, by means of the existential construct 《x≥g》φ, one can enforce that there exist at least g strategies x satisfying φ. Dually, via the universal construct 〚x<g〛φ, one can ensure that all but less than g strategies x satisfy φ.Strategies in GSL are counted semantically. This means that strategies inducing the same outcome, even though looking different, are counted as one. While this interpretation is natural, it requires a suitable machinery to allow for such a counting, as we do. Precisely, we introduce a non-trivial equivalence relation over strategy profiles based on the strategic behavior they induce.To give an evidence of GSL usability, we investigate some basic questions about the Vanilla GSL[1g] fragment, that is the vanilla restriction of the well-studied One-Goal Strategy Logic fragment of SL augmented with graded strategy quantifiers. We show that the model-checking problem for this logic is PTime-complete. We also report on some positive results about the determinacy.

English

The structure of the lattice of clones on a finite set has been proven to be very complex. To better understand the top of this lattice, it is important to provide a characterization of submaximal clones in the lattice of clones. It is known that the clones $Pol(\theta)$ and $Pol(\rho)$ (where $\theta$ is a nontrivial equivalence relation on $E_k = \{0,\dots, k-1\}$, and $\rho$ is among the six types of relations which characterize maximal clones) are maximal clones. In this paper, we provide a classification of relations (of Rosenberg's List) on $E_k$ such that the clone $Pol(\theta) \cap Pol(\rho)$ is maximal in $Pol(\theta)$.

Homogeneous 1‐based structures and interpretability in random structures

Let V be a finite relational vocabulary in which no symbol has arity greater than 2. Let be countable V‐structure which is homogeneous, simple and 1‐based. The first main result says that if is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if is “coordinatized” by a set with SU‐rank 1 and there is no definable (without parameters) nontrivial equivalence relation on M with only finite classes, then is strongly interpretable in a random structure.

Abundance of $E$-order-preserving transformation semigroups

Let T X be the full transformation semigroup on anite totally ordered set X = f 1 < 2 < ::: < ng ( n � 3) and E be a nontrivial equivalence relation on X. In this paper, we consider a subsemigroup of T X dened by EOPX = f f 2 T X : 8 x;y 2 X; ( x;y) 2 E;xy ) ( f ( x) ;f ( y)) 2 E;f ( x) � f ( y) g and present a necessary and sufficient condition under which the semigroup EOPX is abundant.

On the ‘classical’ operations in three-valued logics

The general aim of the present paper is to provide the analysis of the connection between proof-theoretical and functional properties of certain logical matrices. To be more precise, we consider the class of three-valued matrices that induce the classical consequence relation and show that their operations always constitute a subset of one of the maximal classes of functions, which preserve non-trivial equivalence relations. We use a matrix with the single designated value as a sample for in-depth analysis, and generalize the results to suit other cases. Furthermore, on the basis of obtained results we conclude the paper with methodological considerations concerning the nature and interpretation of the truth-values in logical matrices.

Meet-irreducible submaximal clones determined by nontrivial equivalence relations

Let θ and ρ be a nontrivial equivalence relation and a binary relation on a finite set A, respectively. It is known from Rosenberg’s classification theorem (1965) that the clone Pol θ, which consists of all operations on A that preserve θ, is among the maximal clones on A. In the present paper, we find all binary relations ρ such that the clone Pol ρ is a meet-irreducible maximal subclone of Pol θ.

THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION

AbstractLet ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote $$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$ so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.

Binary central relations and submaximal clones determined by nontrivial equivalence relations

Let k ≥ 3, θ a nontrivial equivalence relation on E k = {0, . . . ,k – 1}, and ρ a binary central relation on E k (a reflexive graph with a vertex having E k as its neighborhood). It is known that the clones Pol θ and Pol ρ (of operations on E k preserving θ and ρ, respectively) are maximal clones; i.e., covered by the largest clone in the inclusion-ordered lattice of clones on E k . In this paper, we give the classification of all binary central relations ρ on E k such that the clone Pol θ ∩ Pol ρ is maximal in Pol θ.

Non-commutative field theory approach to two-dimensional vortex liquid system

We investigate the non-commutative (NC) field theory approach to the vortex liquid system restricted to the lowest Landau level (LLL) approximation. NC field theory effectively takes care of the phase space reduction of the LLL physics in a $\star$-product form and introduces a new gauge invariant form of a quartic potential of the order parameter in the Ginzburg-Landau (GL) free energy. This new quartic interaction coupling term has a non-trivial equivalence relation with that obtained by Br\'ezin, Nelson and Thiaville in the usual GL framework. The consequence of the equivalence is discussed.

Definition, Convention, and Simultaneity: Malament's Result and Its Alleged Refutation by Sarkar and Stachel

The question whether distant simultaneity (relativized to an inertial frame) has a factual or a conventional status in special relativity has long been disputed and remains in contention even today. At one point it appeared that Malament (1977) had settled the issue by proving that the only non-trivial equivalence relation definable from (temporally symmetric) causal connectability is the standard simultaneity relation. Recently, however, Sarkar and Stachel (1999) claim to have identified a suspect assumption in the proof by defining a non-standard simultaneity relation from causal connectability. I contend that their critique is based on a misunderstanding of the criteria for the definability of a relation, a misunderstanding that Malement's original treatment helped to foster. There are in fact a variety of notions of definability that can be brought to bear. They all, however, require a condition that suffices to secure Malament's result. The non-standard relation Sarkar and Stachel claim to be definable is not so definable, and, I argue, their proposal to modify the notion of “causal definability” is misguided. Finally, I address the relevance of Malament's result to the thesis of conventionalism.

Transitive primitive permutation groups acting on trees

In this paper we give a classification of certain transitive primitive tree permutation groups (those closed under “patching”), where a transitive tree permutation group ( G, Ω) is primitive if there are no non-trivial equivalence relations on Ω with convex classes which are preserved by each g in G. The classification is obtained by examining the restricted actions of G on the maximal subchains of Ω.

Renormalisation in a circle map with two inflection points

An extended version of the sine circle map becomes non-invertible by developing simultaneously two critical points in part of parameter space. Universality in the corresponding transition from quasiperiodicity to chaos manifests itself in the organisation of doubly superstable orbits. Four operators are needed to generate all such orbits. The existence of non-trivial equivalence relations for operation sequences makes the present theory different from other multi-operator renormalisation approaches. Possible implications of the theory in higher-dimensional systems are discussed.