A copy of this issue can also be found on the
MetaPress server (with DOI names of articles).
Yuzhong Ding.
Several Classes of {BCI}-algebras
and their Properties,
Formalized Mathematics 15(1),
pages 1-9, 2007. MML Identifier: BCIALG_1 Summary: I have formalized the BCI-algebras closely following the book cite{BCIAlgebras},
sections 1.1 to 1.3, 1.6, 2.1 to 2.3, and 2.7. In this article the general theory of {BCI}-algebras
and several classes of {BCI}-algebras are given.
Micha{\l} Trybulec.
Formal Languages -- Concatenation and Closure,
Formalized Mathematics 15(1),
pages 11-15, 2007. MML Identifier: FLANG_1 Summary: Formal languages are introduced as subsets of the set of all 0-based finite
sequences over a given set (the alphabet). Concatenation, the $n$-th power and closure are
defined and their properties are shown. Finally, it is shown that the closure of the alphabet
(understood here as the language of words of length 1) equals to the set of all words over
that alphabet, and that the alphabet is the minimal set with this property. Notation and
terminology were taken from \cite{HOPCROFT-ULLMAN:1979} and \cite{WAITE-GOOS:1984}.
Karol P\c{a}k.
Basic Properties of Determinants
of Square Matrices over a Field,
Formalized Mathematics 15(1),
pages 17-25, 2007. MML Identifier: MATRIX11 Summary: In this paper I present basic properties of the determinant of square
matrices over a field and selected properties of the sign of a permutation. First,
I define the sign of a
permutation by the requirement $${\rm sgn}(p)=\prod_{1\leq i \gt j \leq n}\,{\rm sgn}(p(j)-p(i)),$$
where $p$ is any fixed permutation of a set with $n$ elements. I prove that the sign of
a product of two permutations is the same as the product of their signs and show the relation
between signs and parity of permutations. Then I consider the determinant of a linear combination
of lines, the determinant of a matrix with permutated lines and the determinant of a matrix with
a repeated line. Finally, at the end I prove that the determinant of a product of two square
matrices is equal to the product of their determinants.