### $\mathcal{W}$-completeness and fixpoint properties

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The concept of a $0$-ideal in $0$-distributive posets is introduced. Several properties of $0$-ideals in $0$-distributive posets are established. Further, the interrelationships between $0$-ideals and $\alpha $-ideals in $0$-distributive posets are investigated. Moreover, a characterization of prime ideals to be $0$-ideals in $0$-distributive posets is obtained in terms of non-dense ideals. It is shown that every $0$-ideal of a $0$-distributive meet semilattice is semiprime. Several counterexamples are discussed.

Several characterizations of 0-distributive posets are obtained by using the prime ideals as well as the semiprime ideals. It is also proved that if every proper $l$-filter of a poset is contained in a proper semiprime filter, then it is $0$-distributive. Further, the concept of a semiatom in 0-distributive posets is introduced and characterized in terms of dual atoms and also in terms of maximal annihilator. Moreover, semiatomic 0-distributive posets are defined and characterized. It is shown that...

We characterize totally ordered sets within the class of all ordered sets containing at least four-element chains. We use a simple relationship between their isotone transformations and the so called 1-endomorphism which is introduced in the paper. Later we describe 1-, 2-, 3-, 4-homomorphisms of ordered sets in the language of super strong mappings.

A distributive pseudocomplemented set $S$ [2] is called Stone if for all $a\in S$ the condition $LU({a}^{*},{a}^{**})=S$ holds. It is shown that in a finite case $S$ is Stone iff the join of all distinct minimal prime ideals of $S$ is equal to $S$.

For every countable ordinal α, we construct an ${l}_{1}$-predual ${X}_{\alpha}$ which is isometric to a subspace of $C\left({\omega}^{{\omega}^{{\omega}^{\alpha}+2}}\right)$ and isomorphic to a quotient of $C\left({\omega}^{\omega}\right)$. However, ${X}_{\alpha}$ is not isomorphic to a subspace of $C\left({\omega}^{{\omega}^{\alpha}}\right)$.

We introduce the so-called DN-algebra whose axiomatic system is a common axiomatization of directoids with an antitone involution and the so-called D-quasiring. It generalizes the concept of Newman algebras (introduced by H. Dobbertin) for a common axiomatization of Boolean algebras and Boolean rings.

A complete list of positive Tits-sincere one-peak posets is provided by applying combinatorial algorithms and computer calculations using Maple and Python. The problem whether any square integer matrix $A\in \u2099\left(\mathbb{Z}\right)$ is ℤ-congruent to its transpose ${A}^{tr}$ is also discussed. An affirmative answer is given for the incidence matrices ${C}_{I}$ and the Tits matrices $C{\u0302}_{I}$ of positive one-peak posets I.