# The ordering of commutative terms

• Volume: 56, Issue: 1, page 133-154
• ISSN: 0011-4642

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## Abstract

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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.

## How to cite

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Ježek, Jaroslav. "The ordering of commutative terms." Czechoslovak Mathematical Journal 56.1 (2006): 133-154. <http://eudml.org/doc/31021>.

@article{Ježek2006,
abstract = {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.},
author = {Ježek, Jaroslav},
journal = {Czechoslovak Mathematical Journal},
keywords = {definable; term; definability; term},
language = {eng},
number = {1},
pages = {133-154},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The ordering of commutative terms},
url = {http://eudml.org/doc/31021},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Ježek, Jaroslav
TI - The ordering of commutative terms
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 1
SP - 133
EP - 154
AB - 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.
LA - eng
KW - definable; term; definability; term
UR - http://eudml.org/doc/31021
ER -

## References

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1. The lattice of equational theories. Part I: Modular elements, Czechoslovak Math. J. 31 (1981), 127–152. (1981) MR0604120
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3. The lattice of equational theories. Part III: Definability and automorphisms, Czechoslovak Math. J. 32 (1982), 129–164. (1982) MR0646718
4. The lattice of equational theories. Part IV: Equational theories of finite algebras, Czechoslovak Math. J. 36 (1986), 331–341. (1986) MR0831318
5. Definability in the lattice of equational theories of semigroups, Semigroup Forum 46 (1993), 199–245. (1993) MR1200214
6. 10.1090/S0002-9947-03-03351-8, Trans. Amer. Math. Soc. 356 (2004), 3483–3504. (2004) Zbl1050.08005MR2055743DOI10.1090/S0002-9947-03-03351-8
7. Algebras, Lattices, Varieties, Volume I, Wadsworth & Brooks/Cole, Monterey, CA, 1987. (1987) MR0883644

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