| Title:
|
On subalgebra lattices of a finite unary algebra. II. (English) |
| Author:
|
Pióro, Konrad |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
126 |
| Issue:
|
1 |
| Year:
|
2001 |
| Pages:
|
171-181 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We use graph-algebraic results proved in [8] and some results of the graph theory to characterize all pairs $\langle \mathbf{L}_{1},\mathbf{L}_{2}\rangle $ of lattices for which there is a finite partial unary algebra such that its weak and strong subalgebra lattices are isomorphic to $\mathbf{L}_{1}$ and $\mathbf{L}_{2}$, respectively. Next, we describe other pairs of subalgebra lattices (weak and relative, etc.) of a finite unary algebra. Finally, necessary and sufficient conditions are found for quadruples $\langle \mathbf{L}_{1},\mathbf{L}_{2}, \mathbf{L}_{3},\mathbf{L}_{4}\rangle $ of lattices for which there is a finite unary algebra having its weak, relative, strong subalgebra and initial segment lattices isomorphic to $\mathbf{L}_{1},\mathbf{L}_{2}, \mathbf{L}_{3},\mathbf{L}_{4}$, respectively. (English) |
| Keyword:
|
graph |
| Keyword:
|
finite unary algebra |
| Keyword:
|
partial algebra |
| Keyword:
|
subalgebras |
| Keyword:
|
subalgebra lattices |
| MSC:
|
05C20 |
| MSC:
|
05C40 |
| MSC:
|
05C90 |
| MSC:
|
05C99 |
| MSC:
|
06B15 |
| MSC:
|
06D05 |
| MSC:
|
08A30 |
| MSC:
|
08A55 |
| MSC:
|
08A60 |
| idZBL:
|
Zbl 0978.08004 |
| idMR:
|
MR1826479 |
| DOI:
|
10.21136/MB.2001.133927 |
| . |
| Date available:
|
2009-09-24T21:48:41Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133927 |
| . |
| Reference:
|
[1] W. Bartol: Weak subalgebra lattices.Comment. Math. Univ. Carolin. 31 (1990), 405–410. Zbl 0711.08007, MR 1078473 |
| Reference:
|
[2] W. Bartol, F. Rosselló, L. Rudak: Lectures on Algebras, Equations and Partiality.Rosselló F. (ed.), Technical report B-006, Univ. Illes Balears, Dept. Ciencies Mat. Inf., 1992. |
| Reference:
|
[3] C. Berge: Graphs and Hypergraphs.North-Holland, Amsterdam, 1973. Zbl 0254.05101, MR 0357172 |
| Reference:
|
[4] P. Burmeister: A Model Theoretic Oriented Approach to Partial Algebras.Math. Research Band 32, Akademie Verlag, Berlin, 1986. Zbl 0598.08004, MR 0854861 |
| Reference:
|
[5] P. Crawley, R. P. Dilworth: Algebraic Theory of Lattices.Prentice Hall Inc., Englewood Cliffs, NJ, 1973. |
| Reference:
|
[6] B. Jónsson: Topics in Universal Algebra.Lecture Notes in Mathemathics 250, Springer, Berlin, 1972. MR 0345895 |
| Reference:
|
[7] K. Pióro: On some non-obvious connections between graphs and partial unary algebras.Czechoslovak Math. J. 50 (2000), 295–320. MR 1761388, 10.1023/A:1022418818272 |
| Reference:
|
[8] K. Pióro: On subalgebra lattices of a finite unary algebra, part I.Math. Bohem. 126 (2001), 161–170. MR 1826479 |
| Reference:
|
[9] K. Pióro: On a strong property of the weak subalgebra lattice.Algebra Univers. 40 (1998), 477–495. MR 1681837 |
| Reference:
|
[10] H. E. Robbins: A theorem on graphs with an application to a problem of traffic.Am. Math. Monthly 46 (1939), 281–283. Zbl 0021.35703, MR 1524589, 10.2307/2303897 |
| . |
