This note makes two observations about lattices of subsemilattices. First, we establish relationship between direct decompositions of such lattices and ordinal sum decompositions of semilattices. Then we give a characterization of the subsemilattice-lattices. Let us recall some terminology. L will always stand for a semilattice, whose operation will be denoted by. a of an arbitrary lattice L is...

I re-work Veblen’s ideas to show how hierarchies of normal functions can be generated merely by iterating certain higher order gadgets. As an illustration I show that an application of a Schütte bracket can be evaluated without the need of an intricate recursion.

A model with otp(pcf(a)) = ω1 + 1 is constructed, for countable set a of regular cardinals. 1 Preliminary Settings Let 〈κα | α < ω1〉 be an an increasing continuous sequence of singular cardinals of cofinality ω so that for each α < ω1, if α = 0 or α is a successor ordinal, then κα is a limit of an increasing sequence 〈κα,n | n < ω〉 of cardinals such that (1) κα,n is strong up to a 2-Mahlo cardi...

‎It is shown that‎ ‎if $M$ is an Artinian module over a ring‎ ‎$R$‎, ‎then $M$ has Noetherian dimension $alpha $‎, ‎where $alpha $ is a countable ordinal number‎, ‎if and only if $omega ^{alpha }+2leq it{l}(M)leq omega ^{alpha‎ +1}$, ‎where $ it{l}(M)$ is‎ ‎the length of $M$‎, ‎$i.e.,$ the least ordinal number such that the interval $[0‎, ‎ it{l}(M))$ cannot be embedded in the lattice of all su...

The approach of ordinal mind change complexity, introduced by Freivalds and Smith, uses (notations for) constructive ordinals to bound the number of mind changes made by a learning machine. This approach provides a measure of the extent to which a learning machine has to keep revising its estimate of the number of mind changes it will make before converging to a correct hypothesis for languages...

Canine hip dysplasia (CHD) is a serious and common musculoskeletal disease of pedigree dogs and therefore represents both an important welfare concern and an imperative breeding priority. The typical heritability estimates for radiographic CHD traits suggest that the accuracy of breeding dog selection could be substantially improved by the use of estimated breeding values (EBVs) in place of sel...

We study an extension of Japaridze’s polymodal logic GLP with transfinitely many modalities and develop a provability-algebraic ordinal notation system up to the ordinal Γ0. In the papers [1, 2] a new algebraic approach to the traditional prooftheoretic ordinal analysis was presented based on the concept of graded provability algebra. The graded provability algebra of a formal theory T is its L...

Suppose A is an Abelian p-group and α is an ordinal such that A/pA is a direct sum of countable groups. It is shown that A has a nice basis if, and only if, pA has a nice basis. This strengthens an earlier result of ours in Bull. Allah. Math. Soc. (2011) proved when α < ω.

Rough set theory has been successfully applied in selecting attributes to improve the effectiveness in deriving decision trees/rules for decisions and classification problems. When decisions involve ordinal classes, the rough set reduction process should try to preserve the order relation generated by the decision classes. Previous works on rough sets when applied to ordinal decision systems st...

In this paper we illustrate the potential of ordinal-patterns-based methods for analysis of real-world data and, especially, of electroencephalogram (EEG) data. We apply already known (empirical permutation entropy, ordinal pattern distributions) and new (empirical conditional entropy of ordinal patterns, robust to noise empirical permutation entropy) methods for measuring complexity, segmentat...