Über zweiBirkhoff-Frinksche Struktursätze der allgemeinen Algebra

Diener, Karl-Heinz

doi:10.1007/bf01900683

Cited by 9 publications

(6 citation statements)

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“…Then c < u e C (!) and a < v. Notice that the proofs of Diener [6] and Crawley and Dilworth [5, 2.2] for the weak atomicity of algebraic lattices involve the existence of certain finite (undirected) joins and do not work in the present more general setting.…”

Section: Every Compactly Generated Chain-complete Poset Is Weakly Atomentioning

confidence: 99%

“…Wolk [17]) if every non-empty chain of P has a join and a meet (in other words, if P and its dual P* are up-complete). PROPOSITION 6. Every compactly generated chain-complete poset is weakly atomic.…”

Section: Substructures and Weak Atomicitymentioning

confidence: 99%

“…https://doi.org/10.1017/S1446788700033966 Marcel Ern6 [6] (where F is the Scott operator of P). This map is well-defined because the Scott operator of an algebraic poset is a closure operator.…”

Section: Proof (A) => (B) the Following Map Is An Isomorphism: / : mentioning

confidence: 99%

“…It has been observed by Birkhoff and Frink [3] that every algebraic complete lattice L is weakly atomic. However, the initial proof given in [3] contained a little gap which was closed by K.-H. Diener [6]. A straightforward argument is this: if a < b, then there exists a compact element v of [a, b] (which need not be compact in L) such that a < v. By Zorn's Lemma, we can find a maximal element u with a < u < v. Thus [u, v] is the required jump.…”

Section: Substructures and Weak Atomicitymentioning

confidence: 99%

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Compact generation in partially ordered sets

Erné

1987

J Aust Math Soc A

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Several" "classical" results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own "up-completion" i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.

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Section: Every Compactly Generated Chain-complete Poset Is Weakly Atomentioning

confidence: 99%

Section: Substructures and Weak Atomicitymentioning

confidence: 99%

Section: Proof (A) => (B) the Following Map Is An Isomorphism: / : mentioning

confidence: 99%

Section: Substructures and Weak Atomicitymentioning

confidence: 99%

See 2 more Smart Citations

Compact generation in partially ordered sets

Erné

1987

J Aust Math Soc A

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“…For the description of the properties of si and S£ as quoted in the Theorem, let us remember the following two fundamental composition laws (cf. [18,1], also DIENER [7]) in the complete lattice <P(X) of all filters on X:…”

Section: The Main Questions About These Galois Correspondences Are Anmentioning

confidence: 99%