# Article

Full entry | PDF   (0.3 MB)
Keywords:
unary algebra; congruence lattice; intransitive G-Sets; M-Sets; representations of lattices
Summary:
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.
References:
[1] Burris S., Sankappanavar H.P.: A Course in Universal Algebra. Graduate Texts in Mathematics, 78, Springer, New York-Berlin, 1981; http://www.math.uwaterloo.ca/snburris/htdocs/ualg.html DOI 10.1007/978-1-4613-8130-3 | MR 0648287 | Zbl 0478.08001
[2] Grätzer G., Schmidt E.T.: Characterizations of congruence lattices of abstract algebras. Acta Sci. Math. (Szeged) 24 (1963), 34–59. MR 0151406 | Zbl 0117.26101
[3] McKenzie R.N., McNulty G., Taylor W.: Algebras, Lattices, Varieties, Vol. 1. The Wadsworth & Brooks/Cole Mathematics Series, Monterey, CA, 1987. MR 0883644
[4] Pálfy P.P., Pudlák P.: Congruence lattices of finite algebras and intervals in subgroup lattices in finite groups. Algebra Universalis 11 (1980), 22–27. DOI 10.1007/BF02483080 | MR 0593011
[5] Radeleczki S.: The automorphism group of unary algebras. Math. Pannon. 7 (1996), no. 2, 253–271. MR 1414133 | Zbl 0858.08002
[6] Seif S.: Congruence lattices of algebras–the signed labeling. Proc. Amer. Math. Soc. 124 (1996), no. 5, 1361–1370. DOI 10.1090/S0002-9939-96-03102-4 | MR 1301526
[7] Seif S.: Congruence semimodularity and transitivity-forcing lattices via transitivity labeling. manuscript.
[8] Seif S.: Two-orbit M-Sets and primitive monoids. manuscript.
[9] Seif S.: The probability that a finite lattice has an intransitive G-Set representation. manuscript.
[10] Tůma J.: Intervals in subgroup lattices of infinite groups. J. Algebra 125 (1989), no. 2, 367–399. DOI 10.1016/0021-8693(89)90171-3 | MR 1018952 | Zbl 0679.20024
[11] Vernikov B.M.: On congruence lattices of $G$-sets. Comment. Math. Univ. Carolinae 38 (1997), no. 3, 601–611.

Partner of