Usually, during lectures Turing Machines are firstly introduced from an informal point of view (for example, in this way: http://en.wikipedia.org/wiki/Turing_machine#Informal_description) and then their definition is formalized (for example, in this way: http://en.wikipedia.org/wiki/Turing_machine#Formal_definition).

Is it possible to give another 'equivalent' definition that relies more precisely on algebraic concepts (i.e. algebraic structures: semigroups, monoids, etc; just like, for instance, regular languages are recognized by finite monoids and context-free languages are recognized by a product of a free group and a finite monoid)?