Measurable Riesz spaces

We study measurable elements of a Riesz space $E$, i.e. elements $e \in E \setminus \{0\}$ for which the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ is measurable. In particular, we prove that the set $E_{\rm meas}$ of all measurable elements of a Riesz space $E$ with the principal projection property together with zero is a $\sigma$-ideal of $E$. Another result asserts that, for a Riesz space $E$ with the principal projection property the following assertions are equivalent. (1) The Boolean algebra $\mathcal{U}$ of bands of $E$ is measurable. (2) $E_{\rm meas} = E$ and $E$ satisfies the countable chain condition. (3) $E$ can be embedded as an order dense subspace of $L_0(\mu)$ for some probability measure $\mu$.

A note on generators of the endomorphism semigroup of an infinite countable chain

In this note, we consider the semigroup [Formula: see text] of all order endomorphisms of an infinite chain [Formula: see text] and the subset [Formula: see text] of [Formula: see text] of all transformations [Formula: see text] such that [Formula: see text]. For an infinite countable chain [Formula: see text], we give a necessary and sufficient condition on [Formula: see text] for [Formula: see text] to hold. We also present a sufficient condition on [Formula: see text] for [Formula: see text] to hold, for an arbitrary infinite chain [Formula: see text].

Absolute retractness of automata on directed complete posets

The notion of retractness, which is about having left inverses (reflection) for monomorphisms, is crucial in most branches of mathematics. One very important notion related to it is injectivity, which is about extending morphisms to larger domains and plays a fundamental role in many areas of mathematics as well as in computer science, under the name of complete or partial objects. Absolute retractness is tightly related to injectivity and is in fact equivalent to it in many categories. In this paper, combining the two important notions of actions of semigroups and directed complete posets, which are both crucial abstraction and useful in mathematics as well as in computer science, we consider the category Dcpo-[Formula: see text] of actions of a directed complete semigroup on directed complete posets, and study absolute retractness with respect to both monomorphisms and embeddings in this category. Among other things, we show that absolute retract ([Formula: see text]-)dcpo’s are complete but the converse is not necessarily true. Investigating the converse, we find that if we add the property of being a countable chain to completeness, over some kinds of dcpo-monoids such as dcpo-groups and commutative monoids, we get absolute retractness. Furthermore, we show that there are absolute retract [Formula: see text]-dcpo’s, which are not chains.

A NOTE ON SPACES WITH A RANK 3-DIAGONAL

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ is a space satisfying the discrete countable chain condition with a rank 3-diagonal then the cardinality of $X$ is at most $\mathfrak{c}$.

A NOTE ON SPACES WITH RANK 2-DIAGONAL

We prove that if a space $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ with a rank 2-diagonal either has the countable chain condition or is star countable then the cardinality of $X$ is at most $\mathfrak{c}$.

On countable chains having decidable monadic theory

Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.