An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each ${aₙ}_{n\in \mathbb{N}}conA⁺$ there are ${\lambda ₙ}_{n\in \mathbb{N}}\subseteq (0,\infty )$ and a ∈ A with λₙaₙ ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC ⇒ RUC): if aₙ ↓ 0 then aₙ → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the form D(X) (all extended real continuous functions on the compact space X) showing that (σ) and (OC ⇒ RUC) are equivalent, and equivalent...

Any finitely generated regular variety $𝕍$ of distributive double $p$-algebras is finitely determined, meaning that for some finite cardinal $n\left(𝕍\right)$, any subclass $S\subseteq 𝕍$ of algebras with isomorphic endomorphism monoids has fewer than $n\left(𝕍\right)$ pairwise non-isomorphic members. This result follows from our structural characterization of those finitely generated almost regular varieties which are finitely determined. We conjecture that any finitely generated, finitely determined variety of distributive double $p$-algebras...