This article introduces the free magma M(X) constructed on a set X [6]. Then, we formalize some theorems about M(X): if f is a function from the set X to a magma N, the free magma M(X) has a unique extension of f to a morphism of M(X) into N and every magma is isomorphic to a magma generated by a set X under a set of relators on M(X). In doing it, the article defines the stable subset under the law of composition of a magma, the submagma, the equivalence relation compatible with the law of composition and the equivalence kernel of a function. We also introduce some schemes on the recursive function.
@article{bwmeta1.element.doi-10_2478_v10037-010-0003-0,
author = {Marco Riccardi},
title = {Free Magmas},
journal = {Formalized Mathematics},
volume = {18},
year = {2010},
pages = {17-26},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-010-0003-0}
}
Marco Riccardi. Free Magmas. Formalized Mathematics, Tome 18 (2010) pp. 17-26. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-010-0003-0/
[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. | Zbl 06213858
[2] Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.
[3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
[4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
[5] Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 1990.
[6] Nicolas Bourbaki. Elements of Mathematics. Algebra I. Chapters 1-3. Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989. | Zbl 0673.00001
[7] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
[8] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
[9] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
[10] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
[11] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
[12] Małgorzata Korolkiewicz. Homomorphisms of algebras. Quotient universal algebra. Formalized Mathematics, 4(1):109-113, 1993.
[13] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
[14] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
[15] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.
[16] Andrzej Trybulec. Moore-Smith convergence. Formalized Mathematics, 6(2):213-225, 1997.
[17] Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.
[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
[19] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.
[20] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
[21] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.