Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
2-normal identities; lattices; 2-normalized lattice; 3-level inflation of a lattice
2-normal identities; lattices; 2-normalized lattice; 3-level inflation of a lattice
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 $N_2(L)$.
References:
[1] Wismath, I. Chajda,S. L.: Externalization of lattices. Demonstr. Math (to appear). Zbl 1114.08001
[2] Christie, A., Wang, Q., Wismath, S. L.: Minimal characteristic algebras for $k$-normality. Sci. Math. Jpn. 61 (2005), 547-565. MR 2140115 | Zbl 1080.08001
[3] Chromik, W.: Externally compatible identities of algebras. Demonstr. Math. 23 (1990), 345-355. MR 1101497 | Zbl 0734.08005
[4] Clarke, G. T.: Semigroup varieties of inflations of unions of groups. Semigroup Forum 23 (1981), 311-319. DOI 10.1007/BF02676655 | MR 0638575 | Zbl 0486.20033
[5] Denecke, K., Wismath, S. L.: A characterization of $k$-normal varieties. Algebra Univers. 51 (2004), 395-409. DOI 10.1007/s00012-004-1864-2 | MR 2082134 | Zbl 1080.08002
[6] Denecke, K., Wismath, S. L.: Valuations of terms. Algebra Univers. 50 (2003), 107-128. DOI 10.1007/s00012-003-1824-2 | MR 2026831 | Zbl 1092.08003
[7] Graczyńska, E.: On normal and regular identities. Algebra Univers. 27 (1990), 387-397. DOI 10.1007/BF01190718 | MR 1058483
[8] Graczyńska, E.: Identities and Constructions of Algebras. Opole (2006).
[9] Płonka, J.: P-compatible identities and their applications in classical algebras. Math. Slovaca 40 (1990), 21-30. MR 1094969
Search
Partner of
