TY - JOUR ID - 40448 TI - Choice principles and lift lemmas JO - Categories and General Algebraic Structures with Applications JA - CGASA LA - en SN - 2345-5853 AU - Ern\'e, Marcel AD - Faculty for Mathematics and Physics, IAZD, Leibniz Universit\"at, Welfengarten 1, D 30167 Hannover, Germany. Y1 - 2017 PY - 2017 VL - 6 IS - Speical Issue on the Occasion of Banaschewski's 90th Birthday (I) SP - 121 EP - 146 KW - Choice KW - (super)compact KW - foot KW - free semilattice KW - locale KW - noetherian KW - prime KW - sober KW - well-filtered DO - N2 - We show that in ${\bf ZF}$ set theory without choice, the Ultrafilter Principle (${\bf UP}$) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasicontinuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from ${\bf UP}$ but also from ${\bf DC}$, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets,which is not provable in ${\bf ZF}$ set theory. UR - https://cgasa.sbu.ac.ir/article_40448.html L1 - https://cgasa.sbu.ac.ir/article_40448_f354e76a770fa82f66fa30955e1aba56.pdf ER -