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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

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