Morgan versus Stone: beyond Bool

A Morgan-/Kleene-Stone lattice is a distributive lattices endowed with a dual automorphism negation unary operation whose duplication is subsumed by its lattice partial ordering /``and with its elements exceeded by their negation being also exceeded by its elements exceeding their negation''. This is called [quasi-strong]``a [nearly] Morgan/Kleene lattice'', if its elements [exceededexceeding by their negation] exceed their double negations. [Such subsume the (quasi-)equational join {viz., the (quasi-)variety generated by the union} of the varieties of Morgan and Stone lattices, whose elements are said to be strong, becoming their right algebraic unification-abstraction.] As a matter of fact, certain important algebraic issues, being resolved for Morgan lattices, can be extended to Morgan-Stone ones. This appears to concern that of the lattice of quasi-varieties. More precisely, the one of those of [quasi-]strong Morgan-/Klrrne-Stone lattices appears embedable into the direct product of the distributive non-chain/chain (8/5)-element one of Morgan/Kleene lattices, found in an earlier work of ours, and a (2[+3])-element chain, and so is distributive and has (15/9)[+(14/6)] elements. On the other hand, the one of nearly Morgan/Kleene lattices is proved non-modular and infinite/``76-element with Hasse diagram found and properly depicted here''. And what is more, the present study not merely extends the earlier one beyond Morgan lattices but also provides a complete description of relative subdirect irreducibles of quasi-varieties of quasi-strong/nearly Morgan-Stone/Kleene lattices upon the basis of a universal elaboration concerning profound connections between criticality and relative subdirect irreducibility.