S-acts over a Well-ordered Monoid with Modular Congruence Lattice

This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$–act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$–acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$–acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.

Investigation of insertion tableau evolution in the Robinson-Schensted-Knuth correspondence

Robinson-Schensted-Knuth (RSK) correspondence occurs in different contexts of algebra and combinatorics. Recently, this topic has been actively investigated by many researchers. At the same time, many investigations require conducting the computer experiments involving very large Young tableaux. The article is devoted to such experiments. RSK algorithm establishes a bijection between sequences of elements of linearly ordered set and the pairs of Young tableaux of the same shape called insertion tableau and recording tableau . In this paper we study the dynamics of tableau and the dynamics of different concrete values in tableau during the iterations of RSK algorithm. Particularly, we examine the paths within tableaux called bumping routes along which the elements of an input sequence pass. The results of computer experiments with Young tableaux of sizes up to 108 were presented. These experiments were made using the software package for dealing with 2D and 3D Young diagrams and tableaux.

ON THE SET-THEORETIC STRENGTH OF ELLIS’ THEOREM AND THE EXISTENCE OF FREE IDEMPOTENT ULTRAFILTERS ON ω

Ellis’ Theorem (i.e., “every compact Hausdorff right topological semigroup has an idempotent element”) is known to be proved only under the assumption of the full Axiom of Choice (AC); AC is used in the proof in the disguise of Zorn’s Lemma.In this article, we prove that in ZF, Ellis’ Theorem follows from the Boolean Prime Ideal Theorem (BPI), and hence is strictly weaker than AC in ZF. In fact, we establish that BPI implies the formally stronger (than Ellis’ Theorem) statement “for every family ${\cal A} = \{ ({S_i},{ \cdot _i},{{\cal T}_i}):i \in I\}$ of nontrivial compact Hausdorff right topological semigroups, there exists a function f with domain I such that $f\left( i \right)$ is an idempotent of ${S_i}$, for all $i \in I$”, which in turn implies ACfin (i.e., AC for sets of nonempty finite sets).Furthermore, we prove that in ZFA, the Axiom of Multiple Choice (MC) implies Ellis’ Theorem for abelian semigroups (i.e., “every compact Hausdorff right topological abelian semigroup has an idempotent element”) and that the strictly weaker than MC (in ZFA) principle LW (i.e., “every linearly ordered set can be well-ordered”) implies Ellis’ Theorem for linearly orderable semigroups (i.e., “every compact Hausdorff right topological linearly orderable semigroup has an idempotent element”); thus the latter formally weaker versions of Ellis’ Theorem are strictly weaker than BPI in ZFA. Yet, it is shown that no choice is required in order to prove Ellis’ Theorem for well-orderable semigroups.We also show that each one of the (strictly weaker than AC) statements “the Tychonoff product $2^{\Cal R} $ is compact and Loeb” and $BPI_{\Cal R}$ (BPI for filters on ${\Cal R}$) implies “there exists a free idempotent ultrafilter on ω” (which in turn is not provable in ZF). Moreover, we prove that the latter statement does not imply $BP{I_\omega }$ (BPI for filters on ω) in ZF, hence it does not imply any of $AC_{\Cal R} $ (AC for sets of nonempty sets of reals) and $BPI_{\Cal R} $ in ZF, either.In addition, we prove that the statements “there exists a free ultrafilter on ω”, “there exists a free ultrafilter on ω which is not idempotent”, and “for every IP set $A \subseteq \omega$, there exists a free ultrafilter ${\cal F}$ on ω such that $A \in {\cal F}$” are pairwise equivalent in ZF.

Interval Decomposition Lattices are Balanced

Intervals in binary or n-ary relations or other discrete structures generalize the concept of an interval in a linearly ordered set. They are defined abstractly as closed sets of a closure system on a set, satisfying certain axioms. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions. This result is used to show that the lattice of interval decompositions is balanced, and the case when this lattice is distributive is also characterised.

LINEAR ORDERS REALIZED BY C.E. EQUIVALENCE RELATIONS

LetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$(or equivalently, the relationE)realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure ($\omega /E$; ⊴) is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.

Families of sets which can be represented as sublattices of the lattice of convex subsets of a linearly ordered set

We give necessary and sufficient conditions for a family

MULTIPLE TRANSITIVITY AND MIN-WISE INDEPENDENCE IN PERMUTATION GROUPS

Let Ω be a finite linearly ordered set and let k be a positive integer. A permutation group G on Ω is called co-k-restricted min-wise independent on Ω if [Formula: see text] for any X⊆Ω such that |X|≥|Ω|-k+1 and for any x∈X. We show that co-k-restricted min-wise independent groups are exactly the groups with the property that for each subset X⊆Ω with |X|≤k-1, the stabilizer G{X} of X in G is transitive on Ω\X. Using this fact, we determine all co-k-restricted min-wise independent groups.

The Fuzzy Description Logic ALCFH with Hedge Algebras as Concept Modifiers

In this paper we present the fuzzy description logic ALCFH introduced, where primitive concepts are modified by means of hedges taken from hedge algebras. ALCFH is strictly more expressive than Fuzzy-ALC defined in [11]. We show that given a linearly ordered set of hedges primitive concepts can be modified to any desired degree by prefixing them with appropriate chains of hedges. Furthermore, we define a decision procedure for the unsatisfiability problem in ALCFH, and discuss knowledge base expansion when using terminologies, truth bounds, expressivity as well as complexity issues. We extend [8] by allowing modifiers on non-primitive concepts and extending the satisfiability procedure to handle concept definitions.