Title:

The ordering of commutative terms (English) 
Author:

Ježek, J. 
Language:

English 
Journal:

Czechoslovak Mathematical Journal 
ISSN:

00114642 (print) 
ISSN:

15729141 (online) 
Volume:

56 
Issue:

1 
Year:

2006 
Pages:

133154 
Summary lang:

English 
. 
Category:

math 
. 
Summary:

By a commutative term we mean an element of the free commutative groupoid $F$ of infinite rank. For two commutative terms $a$, $b$ write $a\le b$ if $b$ contains a subterm that is a substitution instance of $a$. With respect to this relation, $F$ is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has no automorphisms except the identity. (English) 
Keyword:

definable 
Keyword:

term 
MSC:

03C40 
MSC:

06A07 
MSC:

08B20 
idZBL:

Zbl 1164.03318 
idMR:

MR2207011 
. 
Date available:

20090924T11:32:04Z 
Last updated:

20200703 
Stable URL:

http://hdl.handle.net/10338.dmlcz/128058 
. 
Reference:

[1] J. Ježek: The lattice of equational theories. Part I: Modular elements.Czechoslovak Math. J. 31 (1981), 127–152. MR 0604120 
Reference:

[2] J. Ježek: The lattice of equational theories. Part II: The lattice of full sets of terms.Czechoslovak Math. J. 31 (1981), 573–603. MR 0631604 
Reference:

[3] J. Ježek: The lattice of equational theories. Part III: Definability and automorphisms.Czechoslovak Math. J. 32 (1982), 129–164. MR 0646718 
Reference:

[4] J. Ježek: The lattice of equational theories. Part IV: Equational theories of finite algebras.Czechoslovak Math. J. 36 (1986), 331–341. MR 0831318 
Reference:

[5] J. Ježek and R. McKenzie: Definability in the lattice of equational theories of semigroups.Semigroup Forum 46 (1993), 199–245. MR 1200214 
Reference:

[6] A. Kisielewicz: Definability in the lattice of equational theories of commutative semigroups.Trans. Amer. Math. Soc. 356 (2004), 3483–3504. Zbl 1050.08005, MR 2055743, 10.1090/S0002994703033518 
Reference:

[7] R. McKenzie, G. McNulty and W. Taylor: Algebras, Lattices, Varieties, Volume I.Wadsworth & Brooks/Cole, Monterey, CA, 1987. MR 0883644 
. 