Analytic composition laws with identities (Q1383863)

In this paper, a loop is considered in its universal algebra form as a set \(Q\) with three binary operations, \(*\), \(/\) and \(\backslash\) and a nullary operation \(e\). The main interest lies in formal analytic loops whose underlying sets are \(N\)-dimensional vector spaces over the set of real formal power series in \(N\) variables with zero constant term. A binary operation is defined on these by means of an \(N\)-tuple \(F(X,Y)=(F^1(X,Y),\dots,F^N(X,Y))\), which satisfies the conditions that \(F^i(X,O)=X^i\) and \(F^i(O,Y)=Y^i\), where \(X=(X^1,\dots,X^N)\), \(Y=(Y^1,\dots,Y^N)\) and \(O=(0,\dots,0)\). Such an operation is shown to have both a left and a right inverse, so we do indeed have a loop. An identity is said to be regular if it is an algebraic consequence of the associative law. Any set of regular identities \(W\) can be turned into an analytic set \(dW\) by expanding the right and left sides and equating terms of equal degree. The interesting result is that sets of identities can be analytically equivalent where they are not algebraically equivalent. For example, the left monoalternative property is analytically (but not algebraically) equivalent to the left alternative identity. An extreme example is the identity \((a*(b*c))*((a*b)*c)=((a*b)*c)*(a*(b*c))\) whose linearization is trivial.