A note on the cardinality of infinite partially ordered sets

Tripled partially ordered sets

In this paper, we introduce tripled partially ordered sets and monotone functions on tripled partiallyordered sets. Some basic properties on these new dened sets are studied and some examples forclarifying are given.

Cardinality for Infinite Sets

How can we determine whether two sets have the same cardinality (or “size”)? The answer to this question, reassuringly, lies in early grade school memories: by demonstrating a pairing between elements of the two sets. More formally, we need to demonstrate a bijection f between the two sets. The bijection sets up a one-to-one correspondence, or pairing, between elements of the two sets. We know ...

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

The relative rank rank(S : U) of a subsemigroup U of a semigroup S is the minimum size of a set V ⊆ S such that U together with V generates the whole of S. As a consequence of a result by Sierpiński it follows that for U ≤ TX , the monoid of all self-maps of an infinite set X, rank(TX : U) is either 0, 1, 2 or uncountable. In this paper we consider the relative ranks rank(TX : OX), where X is a...

A note on partially ordered tree automata

A note on partially ordered generalized terms

In this paper we study the definition of generalized powerset of terms in the context of partially ordered monads. We provide a definition of the powerset of terms over acSLAT and study some possible directions for the definition of a partially ordered term monad.