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...

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...

Using the ordinal-pattern concept in permutation entropy, we propose a model sufficiency test to study given model's point prediction accuracy. Compared with some classical tests, such as Broock et al.'s (1996) test, our proposal does not require sufficient eliminate all structures exhibited estimated residuals. When innovations investigated data's underlying dynamics show certain structure, hi...

We present a probabilistic kernel approach to ordinal regression based on Gaussian processes. A threshold model that generalizes the probit function is used as the likelihood function for ordinal variables. Two inference techniques, based on the Laplace approximation and the expectation propagation algorithm respectively, are derived for hyperparameter learning and model selection. We compare t...