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Remarks on endomorphisms of the closure of implicative semilattices. (Russian) Zbl 0621.06002

An algebra \((L;\wedge,\to,l)\) of type \((2,2,0)\) is called an implicative (Brouwerian) semilattice if \((L;\wedge,l)\) is a meet-semilattice with the greatest element \(l\) and \(\to\) is the operation of a relative pseudocomplement, i.e. \(a\wedge x\leq b\) if and only \(x\leq a\to b\). An endomorphism \(\phi\) of an implicative semilattice \(L\) is said to be closed if \(x\leq x\phi\) and \(x\phi =(x\phi)\phi\) for every \(x\in L\).
The main results:
(1) The closed endomorphisms are characterized, for example, \(\phi\) is a closed endomorphism if and only if \(\phi\) is isotone and \(x\to (y\phi)=(x\to y)\phi\) for every \(x, y\).
(2) The set of all closed endomorphisms forms a distributive lattice (see also C. Tsinakis [Houston J. Math. 5, 427–436 (1979; Zbl 0431.06003)]).
(3) A filter \(F\) is the kernel of some closed endomorphism if and only if \(F\) is comonomial, i.e. every congruence class \(\theta [F]\) possesses a greatest element.

MSC:

06A12 Semilattices
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B10 Lattice ideals, congruence relations
08A35 Automorphisms and endomorphisms of algebraic structures
06D20 Heyting algebras (lattice-theoretic aspects)

Citations:

Zbl 0431.06003
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