Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T13:23:10.654Z Has data issue: false hasContentIssue false

Applications of lattice algebra

Published online by Cambridge University Press:  24 October 2008

Extract

By a finite lattice† is meant any finite class having the following properties:

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1934

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

The terminology and symbols of this paper are borrowed wholesale from the author's article “On the combination of subalgebras”, Proc. Camb. Phil. Soc. 29 (1933), 441464Google Scholar. This will be referred to as “Subalgebras” in later footnotes.

“Subalgebras”, Theorem 10·2, condition (2). By A > B is meant AB but AB.

§ “Subalgebras”, Corollary 6·4.

This result was found by R. Dedekiud, Gesammelte Werke, Braunschweig, 1931, Vol. ii. p. 115.

“Subalgebras”, Theorem 4·1.

“Subalgebras”, Tables I—III.

“Subalgebras”, Theorem 4·1.

§ By the phrase ‘A covers B’ is meant A > B, while A > X > B has no solution. L evidently exists (“Subalgebras”, Theorem 7·1).

“Subalgebras”, p. 446, § 9.

“Subalgebras”, Theorem 10·2.

§ There are really six conditions, because of the symmetry between join and meet, which ascribes to (1)—(3) symmetric counterparts.

Condition (1) can be very considerably weakened. Reference to (2) and VI* permits us to replace ‘AC’ by ‘A is covered by C', and ‘A ∩ (X, C) = (AX, C)’ by ‘A ∩ (X, C) ⊃ (AX, C)’ in the first half, to qualify the second half by demanding that A, B, and C shall all cover the same element E, and to replace the equality by an inequality.

“Subalgebras”, Theorem 26·1.

“Subalgebras”, Theorem 25·2.

§ Cf. Remak, R., “Über minimale invariante Untergruppen, etc.”, Journ. f. Math. 162 (1930), 116.Google Scholar

More precisely, be (1, 1) isomorphic in the sense of van der Waerden, , Moderne Algebra (Berlin, 19301931), 1, pp. 28–9.CrossRefGoogle Scholar

Since coherent normal subgroups are Abelian; see R. Remak, op. cit., Theorem 2.

“Subalgebras”, Theorem 17·4.

“Subalgebras”, p. 460.