| Title:
|
Congruence lattices of intransitive G-Sets and flat M-Sets (English) |
| Author:
|
Seif, Steve |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
54 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
459-484 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
unary algebra |
| Keyword:
|
congruence lattice |
| Keyword:
|
intransitive G-Sets |
| Keyword:
|
M-Sets |
| Keyword:
|
representations of lattices |
| MSC:
|
06B15 |
| MSC:
|
08A30 |
| MSC:
|
08A35 |
| MSC:
|
08A60 |
| idZBL:
|
Zbl 1305.08004 |
| idMR:
|
MR3125070 |
| . |
| Date available:
|
2013-10-01T21:10:51Z |
| Last updated:
|
2016-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143470 |
| . |
| Reference:
|
[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. Zbl 0478.08001, MR 0648287, 10.1007/978-1-4613-8130-3 |
| Reference:
|
[2] Grätzer G., Schmidt E.T.: Characterizations of congruence lattices of abstract algebras.Acta Sci. Math. (Szeged) 24 (1963), 34–59. Zbl 0117.26101, MR 0151406 |
| Reference:
|
[3] McKenzie R.N., McNulty G., Taylor W.: Algebras, Lattices, Varieties, Vol. 1.The Wadsworth & Brooks/Cole Mathematics Series, Monterey, CA, 1987. MR 0883644 |
| Reference:
|
[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. MR 0593011, 10.1007/BF02483080 |
| Reference:
|
[5] Radeleczki S.: The automorphism group of unary algebras.Math. Pannon. 7 (1996), no. 2, 253–271. Zbl 0858.08002, MR 1414133 |
| Reference:
|
[6] Seif S.: Congruence lattices of algebras–the signed labeling.Proc. Amer. Math. Soc. 124 (1996), no. 5, 1361–1370. MR 1301526, 10.1090/S0002-9939-96-03102-4 |
| Reference:
|
[7] Seif S.: Congruence semimodularity and transitivity-forcing lattices via transitivity labeling.manuscript. |
| Reference:
|
[8] Seif S.: Two-orbit M-Sets and primitive monoids.manuscript. |
| Reference:
|
[9] Seif S.: The probability that a finite lattice has an intransitive G-Set representation.manuscript. |
| Reference:
|
[10] Tůma J.: Intervals in subgroup lattices of infinite groups.J. Algebra 125 (1989), no. 2, 367–399. Zbl 0679.20024, MR 1018952, 10.1016/0021-8693(89)90171-3 |
| Reference:
|
[11] Vernikov B.M.: On congruence lattices of $G$-sets.Comment. Math. Univ. Carolinae 38 (1997), no. 3, 601–611. |
| . |
