Geodesic
The ordering of commutative terms
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 133-154

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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.
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.
Classification : 03C40, 06A07, 08B20
Keywords: definable; term 
Ježek, J. The ordering of commutative terms. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 1, pp. 133-154. http://geodesic.mathdoc.fr/item/CMJ_2006_56_1_a9/
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