| Title:
|
2-normalization of lattices (English) |
| Author:
|
Chajda, I. |
| Author:
|
Cheng, W. |
| Author:
|
Wismath, S. L. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
3 |
| Year:
|
2008 |
| Pages:
|
577-593 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\tau $ be a type of algebras. A valuation of terms of type $\tau $ is a function $v$ assigning to each term $t$ of type $\tau $ a value $v(t) \geq 0$. For $k \geq 1$, an identity $s \approx t$ of type $\tau $ is said to be $k$-normal (with respect to valuation $v$) if either $s = t$ or both $s$ and $t$ have value $\geq k$. Taking $k = 1$ with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called $k$-normal (with respect to the valuation $v$) if all its identities are $k$-normal. For any variety $V$, there is a least $k$-normal variety $N_k(V)$ containing $V$, namely the variety determined by the set of all $k$-normal identities of $V$. The concept of $k$-normalization was introduced by K. Denecke and S. L. Wismath in their paper (Algebra Univers., 50, 2003, pp.107-128) and an algebraic characterization of the elements of $N_k(V)$ in terms of the algebras in $V$ was given in (Algebra Univers., 51, 2004, pp. 395--409). In this paper we study the algebras of the variety $N_2(V)$ where $V$ is the type $(2,2)$ variety $L$ of lattices and our valuation is the usual depth valuation of terms. We introduce a construction called the {\it $3$-level inflation} of a lattice, and use the order-theoretic properties of lattices to show that the variety $N_2(L)$ is precisely the class of all $3$-level inflations of lattices. We also produce a finite equational basis for the variety $N_2(L)$. (English) |
| Keyword:
|
2-normal identities |
| Keyword:
|
lattices |
| Keyword:
|
2-normalized lattice |
| Keyword:
|
3-level inflation of a lattice |
| MSC:
|
06B20 |
| MSC:
|
08A40 |
| MSC:
|
08B15 |
| idZBL:
|
Zbl 1174.08003 |
| idMR:
|
MR2455924 |
| . |
| Date available:
|
2010-07-20T13:53:37Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140406 |
| . |
| Reference:
|
[1] Wismath, I. Chajda,S. L.: Externalization of lattices.Demonstr. Math (to appear). Zbl 1114.08001 |
| Reference:
|
[2] Christie, A., Wang, Q., Wismath, S. L.: Minimal characteristic algebras for $k$-normality.Sci. Math. Jpn. 61 (2005), 547-565. Zbl 1080.08001, MR 2140115 |
| Reference:
|
[3] Chromik, W.: Externally compatible identities of algebras.Demonstr. Math. 23 (1990), 345-355. Zbl 0734.08005, MR 1101497 |
| Reference:
|
[4] Clarke, G. T.: Semigroup varieties of inflations of unions of groups.Semigroup Forum 23 (1981), 311-319. Zbl 0486.20033, MR 0638575, 10.1007/BF02676655 |
| Reference:
|
[5] Denecke, K., Wismath, S. L.: A characterization of $k$-normal varieties.Algebra Univers. 51 (2004), 395-409. Zbl 1080.08002, MR 2082134, 10.1007/s00012-004-1864-2 |
| Reference:
|
[6] Denecke, K., Wismath, S. L.: Valuations of terms.Algebra Univers. 50 (2003), 107-128. Zbl 1092.08003, MR 2026831, 10.1007/s00012-003-1824-2 |
| Reference:
|
[7] Graczyńska, E.: On normal and regular identities.Algebra Univers. 27 (1990), 387-397. MR 1058483, 10.1007/BF01190718 |
| Reference:
|
[8] Graczyńska, E.: Identities and Constructions of Algebras.Opole (2006). |
| Reference:
|
[9] Płonka, J.: P-compatible identities and their applications in classical algebras.Math. Slovaca 40 (1990), 21-30. MR 1094969 |
| . |
