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On relatively pseudocomplemented posets and Hilbert algebras. (English) Zbl 0609.06009
A Hilbert algebra is an algebra (A;\(\to,1)\) of type (2,0) satisfying the following axioms: (1) \(a\to 1=1\); (2) \(a\to (b\to a)=1\); (3) (a\(\to (b\to c))\to ((a\to b)\to (a\to c))=1\); (4) \(a\to b=b\to a=1\) implies \(a=b\). The Hilbert algebra structure induces a partial order \(\leq\) on A defined by (5) \(a\leq b\) iff \(a\to b=1\). Evidently, 1 is the greatest element of A. Conversely, every poset with a greatest element can be made into a Hilbert algebra.
The reviewer [Acta. Fac. Rer. Nat. Univ. Comenianae 19, 181-185 (1968; Zbl 0194.325)] has introduced the concept of a relatively pseudocomplemented poset (P;\(\leq,\to)\) as follows: (a]\(\cap (x]\subseteq (b]\) iff (x]\(\subseteq (a\to b]\), where \((u]=\{z\in P:\) \(z\leq u\}\). The author shows that every relatively pseudocomplemented poset (P;\(\leq,\to)\) is a Hilbert algebra, whenever the fundamental operation is choosen to be the relative pseudocomplement. Moreover, the induced order (5) coincides with the original one.
Reviewer: T.Katriňák

06D15 Pseudocomplemented lattices
06A06 Partial orders, general
06D20 Heyting algebras (lattice-theoretic aspects)