Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\langle X\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>X</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> be the free monoid on a generating set X,  and suppose one adjoins to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\langle X\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <mi>X</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> universal 2-sided inverses to a finite set S of its elements. We note an elementary algorithm which yields a normal form for elements of the resulting monoid M.In particular, either M will be the free group on X,  or, more generally, the free product as monoids of the free group on a subset <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X_0\subseteq X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>0</mn> </msub> <mo>⊆</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> with the free monoid on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X\setminus X_0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <msub> <mi>X</mi> <mn>0</mn> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> or M will contain <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$1\!$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mspace width="-0.166667em" /> </mrow> </math></EquationSource> </InlineEquation>-sided invertible elements that are not 2-sided invertible.If S is allowed to be infinite, we show that the corresponding normal form still exists, though it cannot necessarily be computed algorithmically.We note work by others on the related topic of “special monoids”, monoids presented by finitely many generators and finitely many relations of the form <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$w=1.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

Adjoining universal inverses to families of elements of free monoids

期刊论文
出版模式：

开放获取
发表日期：

2026年7月29日

查看全文

George M. Bergman

摘要

Let $\langle X\rangle $ $⟨ X ⟩$ be the free monoid on a generating set X, and suppose one adjoins to $\langle X\rangle $ $⟨ X ⟩$ universal 2-sided inverses to a finite set S of its elements. We note an elementary algorithm which yields a normal form for elements of the resulting monoid M.

In particular, either M will be the free group on X, or, more generally, the free product as monoids of the free group on a subset $X_0\subseteq X$ $X_{0} \subseteq X$ with the free monoid on $X\setminus X_0,$ $X \ X_{0},$ or M will contain $1\!$ $1$ -sided invertible elements that are not 2-sided invertible.

If S is allowed to be infinite, we show that the corresponding normal form still exists, though it cannot necessarily be computed algorithmically.

We note work by others on the related topic of “special monoids”, monoids presented by finitely many generators and finitely many relations of the form $w=1.$ $w = 1 .$