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Diamond identities for relative congruences

Gábor Czédli (1995)

Archivum Mathematicum

For a class K of structures and A ∈ K let C o n * ( A ) resp. C o n K ( A ) denote the lattices of * -congruences resp. K -congruences of A , cf. Weaver [25]. Let C o n * ( K ) : = I { C o n * ( A ) : A ∈ K } where I is the operator of forming isomorphic copies, and C o n r ( K ) : = I { C o n K ( A ) : A ∈ K } . For an ordered algebra A the lattice of order congruences of A is denoted by C o n < ( A ) , and let C o n < ( K ) : = I { C o n < ( A ) : A ∈ K } if K is a class of ordered algebras. The operators of forming subdirect squares and direct products are denoted by Q s and P , respectively. Let λ be a lattice identity and let Σ be a set of lattice identities. Let Σ ⊧ c λ ( r ; Q s , P ) denote...

Direct summands of Goldie extending elements in modular lattices

Rupal Shroff (2022)

Mathematica Bohemica

In this paper some results on direct summands of Goldie extending elements are studied in a modular lattice. An element a of a lattice L with 0 is said to be a Goldie extending element if and only if for every b ≤ a there exists a direct summand c of a such that b ∧ c is essential in both b and c . Some characterizations of decomposition of a Goldie extending element in a modular lattice are obtained.

Distributive lattices with a given skeleton

Joanna Grygiel (2004)

Discussiones Mathematicae - General Algebra and Applications

We present a construction of finite distributive lattices with a given skeleton. In the case of an H-irreducible skeleton K the construction provides all finite distributive lattices based on K, in particular the minimal one.

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