Congruence lattices of intransitive G-Sets and flat M-Sets

Seif, Steve. "Congruence lattices of intransitive G-Sets and flat M-Sets." Commentationes Mathematicae Universitatis Carolinae 54.4 (2013): 459-484. <http://eudml.org/doc/260659>.

@article{Seif2013,
abstract = {An M-Set is a unary algebra $\langle X,M \rangle $ whose set $M$ of operations is a monoid of transformations of $X$; $\langle X,M \rangle $ is a G-Set if $M$ is a group. A lattice $L$ is said to be represented by an M-Set $\langle X,M \rangle $ if the congruence lattice of $\langle X,M \rangle $ is isomorphic to $L$. Given an algebraic lattice $L$, an invariant $\mathbf \{\Pi \}(L)$ is introduced here. $\mathbf \{\Pi \}(L)$ provides substantial information about properties common to all representations of $L$ by intransitive G-Sets. $\mathbf \{\Pi \}(L)$ is a sublattice of $L$ (possibly isomorphic to the trivial lattice), a $\Pi $-product lattice. A $\Pi $-product lattice $\Pi (\lbrace L_i:i\in I\rbrace )$ is determined by a so-called multiset of factors $\lbrace L_i: i\in I\rbrace $. It is proven that if $\mathbf \{\Pi \}(L)\cong \Pi (\lbrace L_i: i\in I\rbrace )$, then whenever $L$ is represented by an intransitive G-Set $\mathbf \{Y\}$, the orbits of $\mathbf \{Y\}$ are in a one-to-one correspondence $\beta $ with the factors of $\mathbf \{\Pi \}(L)$ in such a way that if $|I|> 2$, then for all $i\in I$, $L_\{\beta (i)\}\cong Con (\mathbf \{X\}_i)$; if $|I|=2$, the direct product of the two factors of $\mathbf \{\Pi \}(L)$ is isomorphic to the direct product of the congruence lattices of the two orbits of $\mathbf \{Y\}$. Also, if $\mathbf \{\Pi \}(L)$ is the trivial lattice, then $L$ has no representation by an intransitive G-Set. A second result states that algebraic lattices that have no cover-preserving embedded copy of the six-element lattice $A(1)$ are representable by an intransitive G-Set if and only if they are isomorphic to a $\Pi $-product lattice. All results here pertain to a class of M-Sets that properly contain the G-Sets — the so-called flat M-Sets, those M-Sets whose underlying sets are disjoint unions of transitive subalgebras.},
author = {Seif, Steve},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {unary algebra; congruence lattice; intransitive G-Sets; M-Sets; representations of lattices; unary algebras; congruence lattices; intransitive flat -sets; representations of lattices},
language = {eng},
number = {4},
pages = {459-484},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Congruence lattices of intransitive G-Sets and flat M-Sets},
url = {http://eudml.org/doc/260659},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Seif, Steve
TI - Congruence lattices of intransitive G-Sets and flat M-Sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 4
SP - 459
EP - 484
AB - An M-Set is a unary algebra $\langle X,M \rangle $ whose set $M$ of operations is a monoid of transformations of $X$; $\langle X,M \rangle $ is a G-Set if $M$ is a group. A lattice $L$ is said to be represented by an M-Set $\langle X,M \rangle $ if the congruence lattice of $\langle X,M \rangle $ is isomorphic to $L$. Given an algebraic lattice $L$, an invariant $\mathbf {\Pi }(L)$ is introduced here. $\mathbf {\Pi }(L)$ provides substantial information about properties common to all representations of $L$ by intransitive G-Sets. $\mathbf {\Pi }(L)$ is a sublattice of $L$ (possibly isomorphic to the trivial lattice), a $\Pi $-product lattice. A $\Pi $-product lattice $\Pi (\lbrace L_i:i\in I\rbrace )$ is determined by a so-called multiset of factors $\lbrace L_i: i\in I\rbrace $. It is proven that if $\mathbf {\Pi }(L)\cong \Pi (\lbrace L_i: i\in I\rbrace )$, then whenever $L$ is represented by an intransitive G-Set $\mathbf {Y}$, the orbits of $\mathbf {Y}$ are in a one-to-one correspondence $\beta $ with the factors of $\mathbf {\Pi }(L)$ in such a way that if $|I|> 2$, then for all $i\in I$, $L_{\beta (i)}\cong Con (\mathbf {X}_i)$; if $|I|=2$, the direct product of the two factors of $\mathbf {\Pi }(L)$ is isomorphic to the direct product of the congruence lattices of the two orbits of $\mathbf {Y}$. Also, if $\mathbf {\Pi }(L)$ is the trivial lattice, then $L$ has no representation by an intransitive G-Set. A second result states that algebraic lattices that have no cover-preserving embedded copy of the six-element lattice $A(1)$ are representable by an intransitive G-Set if and only if they are isomorphic to a $\Pi $-product lattice. All results here pertain to a class of M-Sets that properly contain the G-Sets — the so-called flat M-Sets, those M-Sets whose underlying sets are disjoint unions of transitive subalgebras.
LA - eng
KW - unary algebra; congruence lattice; intransitive G-Sets; M-Sets; representations of lattices; unary algebras; congruence lattices; intransitive flat -sets; representations of lattices
UR - http://eudml.org/doc/260659
ER -