On some non-obvious connections between graphs and unary partial algebras

Pióro, Konrad. "On some non-obvious connections between graphs and unary partial algebras." Czechoslovak Mathematical Journal 50.2 (2000): 295-320. <http://eudml.org/doc/30562>.

@article{Pióro2000,
abstract = {In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras $\mathbf \{A\}$ and $\mathbf \{B\}$, their weak subalgebra lattices are isomorphic if and only if their graphs $\{\mathbf \{G\}\}^\{\ast \}(\{\mathbf \{A\}\})$ and $\{\mathbf \{G\}\}^\{\ast \}(\{\mathbf \{B\}\})$ are isomorphic. Secondly, it is shown that for two unary partial algebras $\mathbf \{A\}$ and $\mathbf \{B\}$ if their digraphs $\{\mathbf \{G\}\}(\{\mathbf \{A\}\})$ and $\{\mathbf \{G\}\}(\{\mathbf \{B\}\})$ are isomorphic, then their (weak, relative, strong) subalgebra lattices are also isomorphic. Thirdly, we characterize pairs $<\{\mathbf \{L\}\},\{\mathbf \{A\}\}>$, where $\mathbf \{A\}$ is a unary partial algebra and $\mathbf \{L\}$ is a lattice such that the weak subalgebra lattice of $\mathbf \{A\}$ is isomorphic to $\mathbf \{L\}$.},
author = {Pióro, Konrad},
journal = {Czechoslovak Mathematical Journal},
keywords = {partial unary algebra; isomorphism; lattice of subalgebras},
language = {eng},
number = {2},
pages = {295-320},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On some non-obvious connections between graphs and unary partial algebras},
url = {http://eudml.org/doc/30562},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Pióro, Konrad
TI - On some non-obvious connections between graphs and unary partial algebras
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 2
SP - 295
EP - 320
AB - In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras $\mathbf {A}$ and $\mathbf {B}$, their weak subalgebra lattices are isomorphic if and only if their graphs ${\mathbf {G}}^{\ast }({\mathbf {A}})$ and ${\mathbf {G}}^{\ast }({\mathbf {B}})$ are isomorphic. Secondly, it is shown that for two unary partial algebras $\mathbf {A}$ and $\mathbf {B}$ if their digraphs ${\mathbf {G}}({\mathbf {A}})$ and ${\mathbf {G}}({\mathbf {B}})$ are isomorphic, then their (weak, relative, strong) subalgebra lattices are also isomorphic. Thirdly, we characterize pairs $<{\mathbf {L}},{\mathbf {A}}>$, where $\mathbf {A}$ is a unary partial algebra and $\mathbf {L}$ is a lattice such that the weak subalgebra lattice of $\mathbf {A}$ is isomorphic to $\mathbf {L}$.
LA - eng
KW - partial unary algebra; isomorphism; lattice of subalgebras
UR - http://eudml.org/doc/30562
ER -