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It is convenient to have a general term for referring to alphabets and lexicons, indifferently, without concern for their level of construction.  Therefore, any finite set <math>\underline{\underline{X}}</math> is described as a '''syntactic resource''' for the syntactic domain <math>\underline{X},</math> provided its elements can be used as syntactic primitives to construct the signs and expressions in <math>\underline{X}.</math>  If the primitive signs in a syntactic resource are interpreted to denote primitive objects or primitive operations, then a collection of such objects or operations is described as an ''objective'' or an ''operational'' resource, as the case may be.

It is convenient to have a general term for referring to alphabets and lexicons, indifferently, without concern for their level of construction.  Therefore, any finite set <math>\underline{\underline{X}}</math> is described as a '''syntactic resource''' for the syntactic domain <math>\underline{X},</math> provided its elements can be used as syntactic primitives to construct the signs and expressions in <math>\underline{X}.</math>  If the primitive signs in a syntactic resource are interpreted to denote primitive objects or primitive operations, then a collection of such objects or operations is described as an ''objective'' or an ''operational'' resource, as the case may be.

It is always tempting to seek analogies between formal language theory and algebraic theories, and it is often very useful to do so.  But if one tries to forge an analogy between the relation <math>\underline{\underline{X}} ~\text{is a resource for}~ \underline{X},</math> in the formal language sense, and the relation <math>\underline{\underline{X}} ~\text{is a basis for}~ \underline{X},</math> in the algebraic sense, then it becomes necessary to observe important differences between the two perspectives, as they are currently applied.

It is always tempting to seek analogies between formal languages and algebraic structures, and it is often very useful to do so.  But if one tries to forge an analogy between the relation <math>\underline{\underline{X}} ~\text{is a resource for}~ \underline{X},</math> in the formal language sense, and the relation <math>\underline{\underline{X}} ~\text{is a basis for}~ \underline{X},</math> in the algebraic sense, then it becomes necessary to observe important differences between the two perspectives, as they are currently applied.

 

In formal language theory one typically fixes the syntactic resource <math>\underline{\underline{X}}</math> as the primary reality, that is, as the ruling parameter of discussion, and then considers each formal language <math>\underline{X}</math> that can be generated on <math>\underline{\underline{X}}</math> as a particular subset of the maximal language that is possible on <math>\underline{\underline{X}}.</math>  This direction of approach can be contrasted with what is more usual in algebraic studies, where the generated object <math>\underline{X}</math> is taken as the primary reality, and a basis <math>\underline{\underline{X}}</math> is defined secondarily as a minimal or independent spanning set, but generally serves as only one of many possible bases.

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The linguistic relation "X is a resource for X" is thus exploited in the opposite direction from the relation "X is a basis for X".  There does not seem to be any reason in principle why either study cannot be cast the other way around, but it has to be noted that the current practices, and the preferences that support them, dictate otherwise.

The linguistic relation "X is a resource for X" is thus exploited in the opposite direction from the relation "X is a basis for X".  There does not seem to be any reason in principle why either study cannot be cast the other way around, but it has to be noted that the current practices, and the preferences that support them, dictate otherwise.