Σ -isomorphic algebraic structures

Ivan Chajda; Petr Emanovský

Mathematica Bohemica (1995)

  • Volume: 120, Issue: 1, page 71-81
  • ISSN: 0862-7959

Abstract

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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 properties of Σ .

How to cite

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Chajda, Ivan, and Emanovský, Petr. "$\Sigma $-isomorphic algebraic structures." Mathematica Bohemica 120.1 (1995): 71-81. <http://eudml.org/doc/247804>.

@article{Chajda1995,
abstract = {For an algebraic structure $=(A,F,R)$ or type $$ and a set $\Sigma $ of open formulas of the first order language $L()$ we introduce the concept of $\Sigma $-closed subsets of $$. The set $\mathbb \{C\}_\Sigma ()$ of all $\Sigma $-closed subsets forms a complete lattice. Algebraic structures $$, $$ of type $$ are called $\Sigma $-isomorphic if $\mathbb \{C\}_\Sigma ()\cong \mathbb \{C\}_\Sigma ()$. Examples of such $\Sigma $-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 $\Sigma $-isomorphic algebraic structures in dependence on the properties of $\Sigma $.},
author = {Chajda, Ivan, Emanovský, Petr},
journal = {Mathematica Bohemica},
keywords = {closure system; isomorphism; lattice of $\Sigma $-closed subsets; subalgebras; ideals; algebraic structure; $\Sigma $-closed subset; $\Sigma $-isomorphic structures; closure system; isomorphism; lattice of Sigma-closed subsets; subalgebras; ideals},
language = {eng},
number = {1},
pages = {71-81},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$\Sigma $-isomorphic algebraic structures},
url = {http://eudml.org/doc/247804},
volume = {120},
year = {1995},
}

TY - JOUR
AU - Chajda, Ivan
AU - Emanovský, Petr
TI - $\Sigma $-isomorphic algebraic structures
JO - Mathematica Bohemica
PY - 1995
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 120
IS - 1
SP - 71
EP - 81
AB - For an algebraic structure $=(A,F,R)$ or type $$ and a set $\Sigma $ of open formulas of the first order language $L()$ we introduce the concept of $\Sigma $-closed subsets of $$. The set $\mathbb {C}_\Sigma ()$ of all $\Sigma $-closed subsets forms a complete lattice. Algebraic structures $$, $$ of type $$ are called $\Sigma $-isomorphic if $\mathbb {C}_\Sigma ()\cong \mathbb {C}_\Sigma ()$. Examples of such $\Sigma $-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 $\Sigma $-isomorphic algebraic structures in dependence on the properties of $\Sigma $.
LA - eng
KW - closure system; isomorphism; lattice of $\Sigma $-closed subsets; subalgebras; ideals; algebraic structure; $\Sigma $-closed subset; $\Sigma $-isomorphic structures; closure system; isomorphism; lattice of Sigma-closed subsets; subalgebras; ideals
UR - http://eudml.org/doc/247804
ER -

References

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  8. Maľcev A. I, Algebraic systems, Nauka, Moskva, 1970. (In Russian.) (1970) MR0282908
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  10. Snášel V., λ -lattices, PhD - thesis. Palacky University, Olomouc, 1991. (1991) 
  11. Chajda I., Halas R., Genomorphism of lattices and semilattices, Acta-UPO. To appear. MR1385742

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