On a~semilattice of numberings

V. G. Puzarenko

Matematičeskie trudy, Tome 12 (2009) no. 2, pp. 170-209

Résumé

We study some properties of a $\mathfrak c$-universal semilattice $\mathfrak A$ with the cardinality of the continuum, i.e., of an upper semilattice of $m$-degrees. In particular, it is shown that the quotient semilattice of such a semilattice modulo any countable ideal will be also $\mathfrak c$-universal. In addition, there exists an isomorphism $\imath\colon\mathfrak A\hookrightarrow\mathfrak A$ onto some ideal of the semilattice $\mathfrak A$ such that $\mathfrak A/\imath(\mathfrak A)$ will be also $\mathfrak c$-universal. Furthermore, a property of the group of its automorphisms is obtained. To study properties of this semilattice, the technique and methods of admissible sets are used. More exactly, it is shown that the semilattice $m\Sigma$-degrees $\mathrm L^{\mathbb{HF}(S)}_{m\Sigma}$ on the hereditarily finite superstructure $\mathbb{HF}(S)$ over a countable set $S$ will be a $\mathfrak c$-universal semilattice with the cardinality of the continuum.