On Hilbert algebras generated by the order

On the variety of strong subresiduated lattices

“…The following definition is inspired by the definition of order algebra [3] and the proof of Lemma 3.1, where it was shown that if A is an sr‐lattice chain then $$a\rightarrow b \in \lbrace \Box b,1\rbrace$$ for every $$a,b\in A$$. Definition A $$\Box$$‐sr‐lattice is called a order $$\Box$$‐algebra if for every $$a,b\in A$$ it holds that $$a\rightarrow b \in \lbrace \Box b,1\rbrace$$.…”

“…Order algebras, which are defined as the Hilbert algebras (𝐻, →, 1) which satisfy that 𝑎 → 𝑏 ∈ {𝑏, 1} for every 𝑎, 𝑏 (cf. [2,3] for details), also satisfy that 𝑎 → 𝑏 ∈ {1 → 𝑏, 𝑏} for every 𝑎, 𝑏 since in Hilbert algebras the identity 1 → 𝑏 = 𝑏 is satisfied.…”

“…The equivalence between (1) and ( 2) is immediate, also the equivalence between (3) and ( 4). We will show the equivalence between ( 2) and (3). Assume that the condition ( 2) is satisfied and let 𝑎, 𝑏, 𝑐 ∈ 𝐴.…”

“…">2.Order algebras, which are defined as the Hilbert algebras $$(H,\rightarrow ,1)$$ which satisfy that $$a\rightarrow b\in \lbrace b,1\rbrace$$ for every $$a,b$$ (cf. [2, 3] for details), also satisfy that $$a\rightarrow b \in \lbrace 1\rightarrow b,b\rbrace$$ for every $$a,b$$ since in Hilbert algebras the identity $$1\rightarrow b = b$$ is satisfied. Moreover, in the present paper we give two equational bases for the variety generated by the class of $$\Box$$‐order algebras, which is properly contained in the variety of $$\Box$$‐sr‐lattices and also properly contains the variety of sr‐lattices generated by the class of sr‐lattices whose order is total. Finally, we study some logical aspects of the variety $$\mathrm{S}^{\Box }$$ and the subvariety of $$\Box$$‐sr‐lattices generated by the class of $$\Box$$‐order algebras respectively.…”

On subreducts of subresiduated lattices and logic

“…This implication turns B2 into an order Hilbert algebra with supremum and infimum[2] and hence into a bounded sub-Hilbert lattice 7. We have a → (a ∧ b) = (a → a) ∧ (a → b).…”