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unary algebra; congruence lattice; intransitive G-Sets; M-Sets; representations of lattices
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 $\Pi(\{L_i:i\in I\})$ is determined by a so-called multiset of factors $\{L_i: i\in I\}$. It is proven that if $\mathbf{\Pi}(L)\cong \Pi(\{L_i: i\in I\})$, 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.
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