In the Lewis and Langford formalization of system S1 (1932), besides the deduction rules, the substitution rules are as well used: the uniform substitution and the substitution of strict equivalents. They then obtain systems S2, S3, S4 and S5 adding to the axioms of S1 a new axiom, respectively, without changing the deduction rules. Lemmon (1957) gives a new formalization of systems S1-S5, calling them P1-P5. Is is worthwhile to remark that in the formalization of P2-P5 one does not use any more...
This is a paper about the first attemps of demonstration of the fundamental theorem of algebra.
Before, we analyze the tie between complex numbers and the number of roots of an equation of n-th degree.
In the second paragraph, we see the relation between integration and the fundamental theorem.
Finally, we observe the linear differential equation with constant coefficients and Euler's position about the fundamental theorem, and then we consider d'Alembert's, Euler's...
We presents some relations between the (maximal) spectre of a residuated lattice and the residuated lattices of its regular elements. We note the characterization found for the radical of a residuated lattice via the radical of the residuated lattices of the ragular elements. Finally, this last result is applied in the study of the simplicity and semi-simplicity of a residuated lattice.
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