Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider uniformities associated with a variety of finite monoids V, but  work with arbitrary monoids and not only with free or free profinite monoids. The aim of this paper is to address two general questions on these uniform structures and a few more specialized ones. A first question is whether these uniformities can be defined by a metric or a pseudometric. The second question is the description of continous and uniformly continuous functions.  first give a characterization of these functions in term of recognizable sets and use it to extend a result of Reutenauer and Schutzenberger on continuous functions for the pro-group topology. Next  introduce the notion of hereditary continuity and discuss the behaviour of our three main properties (continuity, uniform continuity, hereditary continuity) under composition, product or exponential. In the last section,  analyze the properties of V-uniform continuity when V is the intersection or the join of a family of varieties and  discuss in some detail the case where V is commutative.
['Jean-Eric Pin', 'Pedro V. Silva']
I am writing a research paper on a idea On profinite uniform structures defined by varieties of finite monoids
[{"from": "human", "value": "I am writing a research paper on a idea On profinite uniform structures defined by varieties of finite monoids"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider uniformities associated with a variety of finite monoids V, but  work with arbitrary monoids and not only with free or free profinite monoids. The aim of this paper is to address two general questions on these uniform structures and a few more specialized ones. A first question is whether these uniformities can be defined by a metric or a pseudometric. The second question is the description of continous and uniformly continuous functions.  first give a characterization of these functions in term of recognizable sets and use it to extend a result of Reutenauer and Schutzenberger on continuous functions for the pro-group topology. Next  introduce the notion of hereditary continuity and discuss the behaviour of our three main properties (continuity, uniform continuity, hereditary continuity) under composition, product or exponential. In the last section,  analyze the properties of V-uniform continuity when V is the intersection or the join of a family of varieties and  discuss in some detail the case where V is commutative."}]