2-normalization of lattices

@article{Chajda2008,
abstract = {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 $v(t) \ge 0$. For $k \ge 1$, 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 $\ge k$. 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 $3$-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 $N_2(L)$.},
author = {Chajda, Ivan, Cheng, W., Wismath, S. L.},
journal = {Czechoslovak Mathematical Journal},
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},
language = {eng},
number = {3},
pages = {577-593},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {2-normalization of lattices},
url = {http://eudml.org/doc/37854},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Chajda, Ivan
AU - Cheng, W.
AU - Wismath, S. L.
TI - 2-normalization of lattices
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 3
SP - 577
EP - 593
AB - 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 $v(t) \ge 0$. For $k \ge 1$, 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 $\ge k$. 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 $3$-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 $N_2(L)$.
LA - eng
KW - 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
UR - http://eudml.org/doc/37854
ER -