Here, we we prove that there is a strictly increasing countable chain of finitary relatively finitely-axiomatizable extensions of (the truth-singular version/extension of) the bounded expansion of first-degree entailments - (TS) BFDE, for short - /``relatively axiomatized by the Modus Ponens rule for material implication'', in which case the chain does not contain its join, and so this, being a finitary extension of (TS) BFDE, is not relatively finitely-axiomatizable. (As a consequence, applying one of our previous works, we immediately get a strictly decreasing chain of finitely-axiomatizable quasi-varieties of bounded De Morgan lattices including the variety of bounded Kleene lattices with non-finitely-axiomatizable intersection. )
Alexej P. Pynko (Wed,) studied this question.