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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.

𝒯 -semiring pairs

Jaiung Jun, Kalina Mincheva, Louis Rowen (2022)

Kybernetika

We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.

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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