A new look at the Jordan-Hölder theorem for semimodular lattices
نویسندگان
چکیده
منابع مشابه
The Jordan-Hölder Theorem
This submission contains theories that lead to a formalization of the proof of the Jordan-Hölder theorem about composition series of finite groups. The theories formalize the notions of isomorphism classes of groups, simple groups, normal series, composition series, maximal normal subgroups. Furthermore, they provide proofs of the second isomorphism theorem for groups, the characterization theo...
متن کاملThe Jordan-Hölder theorem with uniqueness for groups and semimodular lattices
For subnormal subgroups A / B and C / D of a given group G, the factor B/A will be called subnormally down-and-up projective to D/C, if there are subnormal subgroups X /Y such that AY = B, A∩Y = X , CY = D and C∩Y = X . Clearly, B/A ∼= D/C in this case. As G. Grätzer and J.B. Nation [6] have just pointed out, the standard proof of the classical Jordan-Hölder theorem yields somewhat more than wi...
متن کاملAn extension theorem for planar semimodular lattices
We prove that every finite distributive lattice D can be represented as the congruence lattice of a rectangular lattice K in which all congruences are principal. We verify this result in a stronger form as an extension theorem.
متن کاملFrankl's Conjecture for a subclass of semimodular lattices
In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...
متن کاملA Construction of Semimodular Lattices
In this paper we prove that if !.l' is a finite lattice. and r is an integral valued function on !.l' satisfying some very natural then there exists a finite geometric (that is.• semimodular and atomistic) lattice containing asa sublatticesuch that r !.l'restricted to Sf. Moreover. we show that if, for all intervals of. semimodular lattices of length at most r(e) are given. then can be chosen t...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Algebra universalis
سال: 2010
ISSN: 0002-5240,1420-8911
DOI: 10.1007/s00012-011-0104-9