This text includes the definition and basic notions of product of posets, chain-complete and flat posets, flattening operation, and the existence theorems of recursive call using the flattening operator. First part of the article, devoted to product and flat posets has a purely mathematical quality. Definition 3 allows to construct a flat poset from arbitrary non-empty set [12] in order to provide formal apparatus which eanbles to work with recursive calls within the Mizar langauge. To achieve this we extensively use technical Mizar functors like BaseFunc or RecFunc. The remaining part builds the background for information engineering approach for lists, namely recursive call for posets [21].We formalized some facts from Chapter 8 of this book as an introduction to the next two sections where we concentrate on binary product of posets rather than on a more general case.
[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics 1(2):377-382 1990.
[2] Grzegorz Bancerek. Complete lattices. Formalized Mathematics 2(5):719-725 1991.
[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics 1(1):41-46 1990.
[4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics 1(1):91-96 1990.
[5] Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics 6(1):81-91 1997.
[6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics 1(1): 55-65 1990.
[7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics 1(1):153-164 1990.
[8] Czesław Bylinski. Basic functions and operations on functions. Formalized Mathematics 1(1):245-254 1990.
[9] Czesław Bylinski. Partial functions. Formalized Mathematics 1(2):357-367 1990.
[10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics 1(1):47-53 1990.
[11] Agata Darmochwał. Finite sets. Formalized Mathematics 1(1):165-167 1990.
[12] B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press 2002.
[13] Marek Dudzicz. Representation theorem for finite distributive lattices. Formalized Mathematics 9(2):261-264 2001.
[14] Adam Grabowski. On the category of posets. Formalized Mathematics 5(4):501-505 1996.
[15] Kazuhisa Ishida and Yasunari Shidama. Fixpoint theorem for continuous functions on chain-complete posets. Formalized Mathematics 18(1):47-51 2010. doi:10.2478/v10037-010-0006-x.
[16] Artur Korniłowicz. Cartesian products of relations and relational structures. Formalized Mathematics 6(1):145-152 1997.
[17] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics 1(1): 115-122 1990.
[18] Andrzej Trybulec. Tuples projections and Cartesian products. Formalized Mathematics 1(1):97-105 1990.
[19] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics 1(2):387-393 1990.
[20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics 1(1):67-71 1990.
[21] Glynn Winskel. The Formal Semantics of Programming Languages. The MIT Press 1993.
[22] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics 1 (1):73-83 1990.
[23] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics 1(1):181-186 1990.
[24] Mariusz Zynel and Czesław Bylinski. Properties of relational structures posets lattices and maps. Formalized Mathematics 6(1):123-130 1997.