## Summary

We analyse three algorithms: exponentiation by squaring, calculation of maximum, and sorting by exchanging in terms of program algebra over an algebra.

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# Analysis of Algorithms: An Example of a Sort Algorithm

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Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2015: 0.134

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Impact per Publication (IPP) 2015: 0.296

Grzegorz Bancerek

We analyse three algorithms: exponentiation by squaring, calculation of maximum, and sorting by exchanging in terms of program algebra over an algebra.

[1] Grzegorz Bancerek. Mizar analysis of algorithms: Preliminaries.

*Formalized Mathematics*, 15(**3**):87-110, 2007. doi:10.2478/v10037-007-0011-x. [Crossref][2] Grzegorz Bancerek. Program algebra over an algebra.

*Formalized Mathematics*, 20(**4**): 309-341, 2012. doi:10.2478/v10037-012-0037-6. [Crossref][3] Grzegorz Bancerek. Cardinal numbers.

*Formalized Mathematics*, 1(**2**):377-382, 1990.[4] Grzegorz Bancerek. Sorting by exchanging.

*Formalized Mathematics*, 19(**2**):93-102, 2011. doi:10.2478/v10037-011-0015-4. [Crossref][5] Grzegorz Bancerek. Institution of many sorted algebras. Part I: Signature reduct of an algebra.

*Formalized Mathematics*, 6(**2**):279-287, 1997.[6] Grzegorz Bancerek. Complete lattices.

*Formalized Mathematics*, 2(**5**):719-725, 1991.[7] Grzegorz Bancerek. Free term algebras.

*Formalized Mathematics*, 20(**3**):239-256, 2012. doi:10.2478/v10037-012-0029-6. [Crossref][8] Grzegorz Bancerek. Terms over many sorted universal algebra.

*Formalized Mathematics*, 5(**2**):191-198, 1996.[9] Grzegorz Bancerek. The ordinal numbers.

*Formalized Mathematics*, 1(**1**):91-96, 1990.[10] Grzegorz Bancerek. König’s lemma.

*Formalized Mathematics*, 2(**3**):397-402, 1991.[11] Grzegorz Bancerek. Joining of decorated trees.

*Formalized Mathematics*, 4(**1**):77-82, 1993.[12] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps.

*Formalized Mathematics*, 6(**1**):93-107, 1997.[13] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences.

*Formalized Mathematics*, 1(**1**):107-114, 1990.[14] Grzegorz Bancerek and Artur Korniłowicz. Yet another construction of free algebra.

*Formalized Mathematics*, 9(**4**):779-785, 2001.[15] Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions.

*Formalized**Mathematics*, 5(**4**):485-492, 1996.[16] Ewa Burakowska. Subalgebras of many sorted algebra. Lattice of subalgebras.

*Formalized**Mathematics*, 5(**1**):47-54, 1996.[17] Czesław Bylinski. Binary operations.

*Formalized Mathematics*, 1(**1**):175-180, 1990.[18] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets.

*Formalized**Mathematics*, 1(**3**):529-536, 1990.[19] Czesław Bylinski. Functions and their basic properties.

*Formalized Mathematics*, 1(**1**): 55-65, 1990.[20] Czesław Bylinski. Functions from a set to a set.

*Formalized Mathematics*, 1(**1**):153-164, 1990.[21] Czesław Bylinski. Partial functions.

*Formalized Mathematics*, 1(**2**):357-367, 1990.[22] Czesław Bylinski. Galois connections.

*Formalized Mathematics*, 6(**1**):131-143, 1997.[23] Agata Darmochwał. Finite sets.

*Formalized Mathematics*, 1(**1**):165-167, 1990.[24] Małgorzata Korolkiewicz. Homomorphisms of many sorted algebras.

*Formalized Mathematics*, 5(**1**):61-65, 1996.[25] Jarosław Kotowicz, Beata Madras, and Małgorzata Korolkiewicz. Basic notation of universal algebra.

*Formalized Mathematics*, 3(**2**):251-253, 1992.[26] Rafał Kwiatek. Factorial and Newton coefficients.

*Formalized Mathematics*, 1(**5**):887-890, 1990.[27] Takashi Mitsuishi and Grzegorz Bancerek. Lattice of fuzzy sets.

*Formalized Mathematics*, 11(**4**):393-398, 2003.[28] Beata Perkowska. Free many sorted universal algebra.

*Formalized Mathematics*, 5(**1**): 67-74, 1996.[29] Andrzej Trybulec. Binary operations applied to functions.

*Formalized Mathematics*, 1 (**2**):329-334, 1990.[30] Andrzej Trybulec. A scheme for extensions of homomorphisms of many sorted algebras.

*Formalized Mathematics*, 5(**2**):205-209, 1996.[31] Andrzej Trybulec. Many sorted algebras.

*Formalized Mathematics*, 5(**1**):37-42, 1996.[32] Andrzej Trybulec. Many sorted sets.

*Formalized Mathematics*, 4(**1**):15-22, 1993.[33] Michał J. Trybulec. Integers.

*Formalized Mathematics*, 1(**3**):501-505, 1990.[34] Wojciech A. Trybulec. Pigeon hole principle.

*Formalized Mathematics*, 1(**3**):575-579, 1990.[35] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma.

*Formalized**Mathematics*, 1(**2**):387-393, 1990.[36] Zinaida Trybulec. Properties of subsets.

*Formalized Mathematics*, 1(**1**):67-71, 1990.[37] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences.

*Formalized Mathematics*, 9(**4**):825-829, 2001.[38] Edmund Woronowicz. Many argument relations.

*Formalized Mathematics*, 1(**4**):733-737, 1990.[39] Edmund Woronowicz. Relations and their basic properties.

*Formalized Mathematics*, 1 (**1**):73-83, 1990.

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Published in Print: 2013-01-01Citation Information:Formalized Mathematics. Volume 21, Issue 1, Pages 1–23, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/forma-2013-0001, January 2013This content is open access.