Around a neutral element in a nearlattice
Let V be a variety with two distinct nullary operations 0 and 1. An algebra 𝔄 ∈ V is called balanced if for each Φ,Ψ ∈ Con(𝔄), we have [0]Φ = [0]Ψ if and only if [1]Φ = [1]Ψ. The variety V is called balanced if every 𝔄 ∈ V is balanced. In this paper, balanced varieties are characterized by a Mal'cev condition (Theorem 3). Furthermore, some special results are given for varieties of bounded lattices.
We characterize d-lattices as those bounded lattices in which every maximal filter/ideal is prime, and we show that a d-lattice is complemented iff it is balanced iff all prime filters/ideals are maximal.
We prove the following theorem: Given a⊆ω and , if for some and all u ∈ WO of length η, a is , then a is .We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: -Turing-determinacy implies the existence of .
A constructively valid counterpart to Bourbaki’s Fixpoint Lemma for chain-complete partially ordered sets is presented to obtain a condition for one closure system in a complete lattice to be stable under another closure operator of . This is then used to deal with coproducts and other aspects of frames.