Bulletin of the Section of Logic
Volume 14/2 (1985), pp. 52–55
reedition 2005 [original edition, pp. 52–56]
Jānis Cirulis
ON CLOSURE ENDOMORPHISMS OF IMPLICATIVE
SEMILATTICES
We present here, without proofs, some results from a paper which will
appear (in Russian) in Latvijskij Matematičeckij Ezegodnik.
Let L be an implicative semilattice, i.e. an algebra (L, ∧, →, 1) such
that (L, ∧) is a meet semilattice, 1 is the greatest element in L, and → is
an operation of pseudocomplementation defined by
a → b = max{x : a ∧ x ≤ b}.
A mapping ϕ : L → L may satisfy any, or none, of the following conditions:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
ϕ(a → b) = ϕa → ϕb,
ϕ(a ∧ b) = ϕa ∧ ϕb,
a ≤ ϕb ⇔ ϕa ≤ ϕb,
ϕa → ϕb = a → ϕb,
a → ϕb = ϕ(a → b),
a = ϕa ∧ d for some d ∈ ϕ−1 (1),
ϕa = c → a for some c.
ϕ is a closure operator on L, if it satisfies (3). If, in addition, it satisfies
also (1) and (2), i.e. is an endomorphism of L, it is said to be a closure
endomorphism (cf. [3]). For example, each the following mappings, with p
arbitrary, is a closure endomorphism of L:
αp a = p → a,
βp a = (a → p) → a.
On Closure Endomorphisms of Implicative Semilattices
53
Theorem 1. The following propositions are equivalent:
(a) ϕ is idempotent, expansive (i.e. a ≤ ϕa for all a) and satisfies (1),
(b) any two of identities (1), (4), (5) hold,
(c) ϕ is isotonic and satisfies (5),
(d) ϕ is multiplicative (2), and anyone of the conditions (5), (6), (7) is
fulfilled,
(e) ϕ satisfies (4) and (7).
The set CE of all closure endomorphisms is closed with respect to the
composition ◦, as well as the pointwise defined meet operation ∧. In [3], it
is shown by direct calculations that the algebra (CE, ◦, ∧) is a distributive
lattice. Our next theorem shows that this result is a simple consequence of
the fact that the filters of a semilattice form a distributive lattice. Recall
that a filter J is called comonomial [1], if every equivalence class modulo
J has the greatest element.
Theorem 2. Let Kϕ be the kernel filter of the endomorphism ϕ. Then
the transformation K : ϕ 7→ Kϕ is an embedding of the algebra CE into
the filter lattice of L, and
ϕa = max{x : x → a ∈ Kϕ }.
The sublattice K(CE) consists just of all comonomial filters. Moreover,
Kαp = p for all p.
A somewhat similar theorem holds also for fixed point sets of closure
endomorphisms. We call a subset A ⊂ L special if for any pair (a, b) ∈ L×A
there is some c ∈ L such that c → a ∈ A and c → b = b. Every special
subset A of L is total in the sense of [2]: b ∈ A implies a → b ∈ A for
any a. The special subalgebras of L form a sublattice of the lattice of all
subalgebras.
Theorem 3. Let Fϕ be the set of fixed points of the closure operator ϕ.
Then the transformation F : ϕ 7→ Fϕ is an anti-isomorphism of CE into
the lattice of special subalgebras of L, and
ϕa = min{x → a ∈ Fϕ : x ∈ L}.
54
Jānis Cirulis
Therefore, for any ϕ ∈ CE Fϕ is really a subalgebra of L known as a
fixed algebra of ϕ. We do not know, however, any proper characteristic of
those special subalgebras corresponding to closure endomorphisms. Nevertheless, the fixed algebras of closure endomorphisms may be characterized
as closure retracts of a certain kind. Recall that a closure retract in L is a
subset R ⊂ L such that for every a ∈ L a relativized filter {x ∈ R : a ≤ x}
has the smallest element [4]. The range of any closure operator is a closure
retract, and this correspondence between closure operators and closure retracts is one-to-one. Now, a closure retract is a fixed algebra of some closure
endomorphism iff it is special in the above sense. Put it in another way, a
special subset is a fixed algebra of a closure endomorphism iff it is a closure
retract.
We say that the implication operation in L is Boolean, if for every
a, b (a → b) → a = a. Equivalently, → is Boolean if (a → b) → b =
(b → a) → a. In this case L becomes a lattice, and a ∨ b = (a → b) → b.
Moreover, the mapping
γp a = p ∨ a
is then a closure endomorphism for every p.
Theorem 4. If the implication in L is Boolean, then
(a) for all ϕ ∈ CE, ϕ(a ∨ b) = ϕa ∨ ϕb = ϕa ∨ b,
(b) the join operation in CE is pointwise: (ϕ ◦ ψ)a = ϕa ∨ ψa,
(c) {αp : p ∈ L} is an ideal in CE,
(d) {γp : p ∈ L} is filter in CE.
References
[1] T. Katrinǎk, Die Kennzeichung der distributiven pseudokomplementären Halbverbände, Journal für die reine und angewandte Mathematic 241 (1970), pp.160–179.
[2] W. C. Nemitz, Implicative homomorphisms with finite ranges,
Proceedings of American Mathematical Society 33 (1972), pp. 319–
322.
[3] C. Tsinakis, Brouwerian semilattices determined by their endomorphism semigroups, Houston Journal of Mathematics 5 (1979), pp. 427–
436.
On Closure Endomorphisms of Implicative Semilattices
55
[4] J. Schmidt, C. Tsinakis, Relative pseudo-complementations joinextensions, and meet-retractions, Mathematishe Zeitschrift 15 (1977),
pp. 271–284.
Department of Physics and Mathematics
Latvian State University
Riga, Latvia (USSR)