Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
See all formats and pricing
More options …
Volume 20, Issue 4


Program Algebra over an Algebra

Grzegorz Bancerek
  • Faculty of Computer Science ,Białystok Technical University, Wiejska 45A, 15-351 Białystok, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2013-02-02 | DOI: https://doi.org/10.2478/v10037-012-0037-6


We introduce an algebra with free variables, an algebra with undefined values, a program algebra over a term algebra, an algebra with integers, and an algebra with arrays. Program algebra is defined as universal algebra with assignments. Programs depend on the set of generators with supporting variables and supporting terms which determine the value of free variables in the next state. The execution of a program is changing state according to successor function using supporting terms.

  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Google Scholar

  • [2] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421-427, 1990.Google Scholar

  • [3] Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics, 1(3):589-593, 1990.Google Scholar

  • [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [5] Grzegorz Bancerek. Cartesian product of functions. Formalized Mathematics, 2(4):547-552, 1991.Google Scholar

  • [6] Grzegorz Bancerek. K¨onig’s lemma. Formalized Mathematics, 2(3):397-402, 1991.Google Scholar

  • [7] Grzegorz Bancerek. Algebra of morphisms. Formalized Mathematics, 6(2):303-310, 1997.Google Scholar

  • [8] Grzegorz Bancerek. Institution of many sorted algebras. Part I: Signature reduct of an algebra. Formalized Mathematics, 6(2):279-287, 1997.Google Scholar

  • [9] Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries. Formalized Mathematics, 15(3):87-110, 2007, doi:10.2478/v10037-007-0011-x.CrossrefGoogle Scholar

  • [10] Grzegorz Bancerek. Sorting by exchanging. Formalized Mathematics, 19(2):93-102, 2011, doi: 10.2478/v10037-011-0015-4.CrossrefGoogle Scholar

  • [11] Grzegorz Bancerek. Free term algebras. Formalized Mathematics, 20(3):239-256, 2012, doi: 10.2478/v10037-012-0029-6.CrossrefGoogle Scholar

  • [12] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Google Scholar

  • [13] Grzegorz Bancerek and Piotr Rudnicki. The set of primitive recursive functions. FormalizedMathematics, 9(4):705-720, 2001.Google Scholar

  • [14] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. FormalizedMathematics, 5(4):485-492, 1996.Google Scholar

  • [15] Ewa Burakowska. Subalgebras of the universal algebra. Lattices of subalgebras. FormalizedMathematics, 4(1):23-27, 1993.Google Scholar

  • [16] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Google Scholar

  • [17] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. FormalizedMathematics, 1(3):529-536, 1990.Google Scholar

  • [18] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Google Scholar

  • [19] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [20] Czesław Bylinski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.Google Scholar

  • [21] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Google Scholar

  • [22] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Google Scholar

  • [23] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Google Scholar

  • [24] Artur Korniłowicz. On the group of automorphisms of universal algebra & many sorted algebra. Formalized Mathematics, 5(2):221-226, 1996.Google Scholar

  • [25] Artur Korniłowicz and Marco Riccardi. The Borsuk-Ulam theorem. Formalized Mathematics, 20(2):105-112, 2012, doi: 10.2478/v10037-012-0014-0.CrossrefWeb of ScienceGoogle Scholar

  • [26] Małgorzata Korolkiewicz. Homomorphisms of many sorted algebras. Formalized Mathematics, 5(1):61-65, 1996.Google Scholar

  • [27] Jarosław Kotowicz, Beata Madras, and Małgorzata Korolkiewicz. Basic notation of universal algebra. Formalized Mathematics, 3(2):251-253, 1992.Google Scholar

  • [28] Yatsuka Nakamura and Grzegorz Bancerek. Combining of circuits. Formalized Mathematics, 5(2):283-295, 1996.Google Scholar

  • [29] Andrzej Nedzusiak. Probability. Formalized Mathematics, 1(4):745-749, 1990.Google Scholar

  • [30] Beata Perkowska. Free many sorted universal algebra. Formalized Mathematics, 5(1):67-74, 1996.Google Scholar

  • [31] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Google Scholar

  • [32] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Google Scholar

  • [33] Andrzej Trybulec. Many sorted sets. Formalized Mathematics, 4(1):15-22, 1993.Google Scholar

  • [34] Andrzej Trybulec. Many sorted algebras. Formalized Mathematics, 5(1):37-42, 1996.Google Scholar

  • [35] Andrzej Trybulec. A scheme for extensions of homomorphisms of many sorted algebras. Formalized Mathematics, 5(2):205-209, 1996.Google Scholar

  • [36] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Google Scholar

  • [37] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Google Scholar

  • [38] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.Google Scholar

  • [39] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [40] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Google Scholar

  • [41] Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 1990.Google Scholar

  • [42] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990. Google Scholar

About the article

This work has been supported by the Polish Ministry of Science and Higher Education project “Managing a Large Repository of Computer-verified Mathematical Knowledge” (N N519 385136).

Published Online: 2013-02-02

Published in Print: 2012-12-01

Citation Information: Formalized Mathematics, Volume 20, Issue 4, Pages 309–341, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-012-0037-6.

Export Citation

This content is open access.

Comments (0)

Please log in or register to comment.
Log in