In this paper new concepts countable join property, countable meet property, P–lattice and Pδ –lattice are introduced. We established that P–lattice and Pδ –lattice are measureable partial lattices and characterized partial lattices of a lattice through countable join and meet properties. We also established some interesting result on the injective property of the lattice measurable functio...

This development defines a well-ordered type of countable ordinals. It includes notions of continuous and normal functions, recursively defined functions over ordinals, least fixed-points, and derivatives. Much of ordinal arithmetic is formalized, including exponentials and logarithms. The development concludes with formalizations of Cantor Normal Form and Veblen hierarchies over normal functions.

The purpose of this paper is to contribute the theory profinite semigroups by considering special class consisting those all whose finitely generated closed subsemigroups are countable, which said be locally countable. We also call countable a pseudovariety V (of finite semigroups) for pro-V investigate operations preserving local countability pseudovarieties and show that, in contrast with fin...