<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">In this article, we defined complex curve and complex integral. Then we have proved the linearity for the complex integral. Furthermore, we have proved complex integral of complex curve's connection is the sum of each complex integral of individual complex curve.</abstract>

Complex Integral

<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">The following theorem is due to Dilworth [8]: Let <italic>P</italic> be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of <italic>P</italic> is <italic>k</italic>, then <italic>P</italic> is the union of <italic>k</italic> chains (cliques).In this article we formalize an elegant proof of the above theorem for finite posets by Perles [13]. The result is then used in proving the case of infinite posets following the original proof of Dilworth [8].A dual of Dilworth's theorem also holds: a poset with maximum clique <italic>m</italic> is a union of <italic>m</italic> independent sets. The proof of this dual fact is considerably easier; we follow the proof by Mirsky [11]. Mirsky states also a corollary that a poset of <italic>r</italic> x <italic>s</italic> + 1 elements possesses a clique of size <italic>r</italic> + 1 or an independent set of size <italic>s</italic> + 1, or both. This corollary is then used to prove the result of Erdős and Szekeres [9].Instead of using posets, we drop reflexivity and state the facts about anti-symmetric and transitive relations.</abstract>

Dilworth's Decomposition Theorem for Posets

<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">In this article we present definitions, basic properties and some examples of periodic functions according to [5].</abstract>

Basic Properties of Periodic Functions

<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">An epsilon number is a transfinite number which is a fixed point of an exponential map: ωϵ = ϵ. The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, ↑). Namely, the ordinal indexing of epsilon numbers is defined as follows:<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2009.17.issue-4_figure1.gif"></inline-graphic>and for limit ordinal λ:<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2009.17.issue-4_figure2.gif"></inline-graphic>Tetration stabilizes at ω:<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2009.17.issue-4_figure3.gif"></inline-graphic>Every ordinal number α can be uniquely written as<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2009.17.issue-4_figure4.gif"></inline-graphic>where κ is a natural number, n1, n2, …, nk are positive integers, and β1 > β2 > … > βκ are ordinal numbers (βκ = 0). This decomposition of α is called the Cantor Normal Form of α.</abstract>

Epsilon Numbers and Cantor Normal Form

<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">Rough sets, developed by Pawlak [6], are an important tool to describe a situation of incomplete or partially unknown information. One of the algebraic models deals with the pair of the upper and the lower approximation. Although usually the tolerance or the equivalence relation is taken into account when considering a rough set, here we rather concentrate on the model with the pair of two definable sets, hence we are close to the notion of an interval set. In this article, the lattices of rough sets and intervals are formalized. This paper, being essentially the continuation of [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].</abstract>

On the Lattice of Intervals and Rough Sets

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