Narrow Results

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let $(X,\le ,c)$ be a finite poset, where $c:X\to X$ is an order-reversing and involutive map such that $c(x)\ne x$ for each $x\in X$. Let $B_2=\{N<P\}$ be the Boolean lattice with two elements and ${{𝒲}}_{+}(X,B_2)$ the family of all the order-preserving 2-valued maps $A:X\to B_2$ such that $A(c(x))=P$ if $A(x)=N$ for all $x\in X$. In this paper, we build a family ${{ℬ}}_{w+}(X)$ of particular subsets of $X$, that we call $w_{+}$-bases on $X$, and we determine a bijection between the family ${{ℬ}}_{w+}(X)$ and the family ${{𝒲}}_{+}(X,B_2)$. In such a bijection, a $w_{+}$-basis $\mathrm{\Omega }$ on $X$ corresponds to a map $A\in {{𝒲}}_{+}(X,B_2)$ whose restriction of $A$ to $\mathrm{\Omega }$ is the smallest 2-valued partial map on $X$ which has $A$ as its unique extension in ${{𝒲}}_{+}(X,B_2)$. Next we show how each $w_{+}$-basis on $X$ becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

Several structural properties of an algebraic structure can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In this paper, we exhibit finite representations of this sequence for finite algebras from congruence modular varieties.

Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term $a$ which is absorbent for addition. We study the problem of enumerating all finite common meadows of order n (that is, common meadows with n elements). This problem turns out to be deeply connected with both the number of finite rings of order n and with the number of a certain kind of partition of positive integers.

In this paper, we showed that any clone of operations preserving a nontrivial $n$-equivalence on a finite set is categorically equivalent to the clone of operations preserving a nontrivial $n$-equivalence on a set having the cardinality of $3$.