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( L , ϕ ) -representations of algebras

Andrzej Walendziak (1993)

Archivum Mathematicum

In this paper we introduce the concept of an ( L , ϕ ) -representation of an algebra A which is a common generalization of subdirect, full subdirect and weak direct representation of A . Here we characterize such representations in terms of congruence relations.

2-normalization of lattices

Ivan Chajda, W. Cheng, S. L. Wismath (2008)

Czechoslovak Mathematical Journal

Let τ be a type of algebras. A valuation of terms of type τ is a function v assigning to each term t of type τ a value v ( t ) ≥ 0 . For k ≥ 1 , an identity s ≈ t of type τ is said to be k -normal (with respect to valuation v ) if either s = t or both s and t have value ≥ k . Taking k = 1 with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called k -normal (with respect to the valuation v ) if all its identities are k -normal. For any variety V , there is a least...

Σ -Hamiltonian and Σ -regular algebraic structures

Ivan Chajda, Petr Emanovský (1996)

Mathematica Bohemica

The concept of a -closed subset was introduced in [1] for an algebraic structure = ( A , F , R ) of type and a set of open formulas of the first order language L ( ) . The set C ( ) of all -closed subsets of forms a complete lattice whose properties were investigated in [1] and [2]. An algebraic structure is called - hamiltonian, if every non-empty -closed subset of is a class (block) of some congruence on ; is called - regular, if = 𝔽 for every two , 𝔽 ∈ whenever they have a congruence class B ∈ C ( ) in common....

Σ -isomorphic algebraic structures

Ivan Chajda, Petr Emanovský (1995)

Mathematica Bohemica

For an algebraic structure = ( A , F , R ) or type and a set Σ of open formulas of the first order language L ( ) we introduce the concept of Σ -closed subsets of . The set ℂ Σ ( ) of all Σ -closed subsets forms a complete lattice. Algebraic structures , of type are called Σ -isomorphic if ℂ Σ ( ) ≅ ℂ Σ ( ) . Examples of such Σ -closed subsets are e.g. subalgebras of an algebra, ideals of a ring, ideals of a lattice, convex subsets of an ordered or quasiordered set etc. We study Σ -isomorphic algebraic structures in dependence on the...

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