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PUP SCITECH R & D CENTER

A SOFTWARE FOR THE CONSTRUCTION. AND ANALYSIS OF FINITE ALGEBRAS. F I N I T A S. Sta. Mesa, Manila, Philippines . PUP SCITECH R & D CENTER. Raoul E. Cawagas. Polytechnic University of the Philippines Manila, Philippines. ABOUT F I N I T A S.

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PUP SCITECH R & D CENTER

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  1. A SOFTWARE FOR THE CONSTRUCTION AND ANALYSIS OF FINITE ALGEBRAS F I N I T A S Sta. Mesa, Manila, Philippines PUP SCITECH R & D CENTER Raoul E. Cawagas Polytechnic University of the Philippines Manila, Philippines

  2. ABOUT F I N I T A S FINITAS is a unique finite algebra package designed for the construction and analysis of finite algebraic structures like groupoids, quasigroups, loops (which are not associative) as well as semigroups and groups (which are associative). FINITAS is user friendly. You do not need to learn special syntax in order to use it. Construction and analysis of a finite system is based on the structure matrix methodin which a system is defined in terms of its Cayley table. This is in marked contrast to most mathematical software in the market today.

  3. THE NEED FOR FINITAS • The study of finite algebras (like quasigroups, loops, semigroups and groups) requires powerful computer algorithms for solving problems involving their construction and analysis. • Most softwares available today can deal only with associative algebras like groups and related structures. For non-associative algebras like quasigroups and NAFIL loops, there appears to be no available softwares. Thus, there is a need for special softwares like FINITAS.

  4. SOME APPLICATIONS • FINITAS is an effective research tool in the field of finite algebras. It has proven to be a very powerful computational tool that has led to various significant discoveries in the development of the theoryof Non-Associative FiniteInvertible Loops (NAFIL) and also in the more general field of quasigroup theory. • FINITAS has also been found to be a very effective mathematics learning aid. Using the software, a college or high school student can easily create his own finite algebra, analyze it to determine its properties and relate it to other known systems within a typical class hour.

  5. HOW TO RUNFINITAS FINITAS runs under Windows. So, first open your computer under Windows. • The package consists of several files: Finitas.exe, Finitas.dsk, FinHelp.chm, Fas.ico, Sample Tables and Sample Blocks. • To open it, simply click on the FINITAS icon. This will bring out the FINITAS title page labeled: Welcome to FINITAS. • Then click on the Ok button to bring out a blank table editor window labeled UNTITLED.TBL. • For detailed instructions, open the Help menu.

  6. FINITASTITLE PAGE This is the FINITAS main screen with its title page labeled Welcome to FINITAS.

  7. TABLE EDITOR WINDOW The table editor window is the main work area of FINITAS . It is basically a blank Cayley table used for constructing a finite algebra.

  8. HOW TO CONSTRUCT AN ALGEBRA A finite algebraic system consists of a set G of finite ordern and a binary operation satisfying a set of Axioms. Such a system is usually denoted by (G,). • A finite algebra is completely determined by the n2 possible binary products of its n elements arranged in the form of a multiplication table called a Cayley table. Thus, to construct a finite algebra we simply construct its Cayley table. • To do this, we first define the set G, the operation , and the Axioms it is required to satisfy. Then we define the n2binary products of its elements and tabulate them in the form of a Cayley table.

  9. Algebraic Axioms Most algebraic structures satisfy some or all of the following Axioms (or postulates): Let S = {1,2,3,…,n} be a set of order n and let  be a binary operation on the set S. • A1 (Closure Axiom): For all a,b  S, ab  S. • A2 (Identity Axiom): There exists an element e  S such that ea = a = ae for all a  S. • A3 (Inverse Axiom): Given an identity e  S, to each a  S there is a unique a’ S such that aa’ = e = a’a. • A4 (Unique Solution Axiom): For all all a,b  S, there exists unique x,y S such that ax = b and ya = b. • A5 (Commutative Axiom): For all a,b  S, ab = ba. • A6 (Associative Axiom): For all a,b  S, a(b c) = (a b)c.

  10. Construction Procedure • First, decide what kind of an algebraic system (G,) you want to construct. This is determined primarily by the Axioms it is required to satisfy. • Next, select your mode of construction: MANUAL or SEMI-AUTO. • The MANUAL mode is the default. If you choose this, you can enter your data directly into the blank Cayley table in the Table Editor window. • If you choose the SEMI-AUTO mode, first open the Construct menu and select the item Parameters to bring out the Parameters dialog box. Here, you must indicate the order n of the system and the Axioms it is required to satisfy.

  11. PARAMETERS DIALOG BOX for construction under the SEMI-AUTO mode. This box is used to indicate the order n and the Axioms of an algebra under construction. To select a given Axiom, you must activate it.

  12. Sample Construction Let us construct an abelian loop (G, ) of order n = 6 using the SEMI-AUTO mode. Here, we take G to be the set G = {1,2,3,4,5,6}. The subset {1,2,3} must form a subsystem of order m = 3. • In the Parameters dialog box, we enter 6 for order n, activate all of the Axioms: A1, A3, A4, A5, and enter the numeral 1 for A2 to serve as the identity element. See Figure G-P. • Then click the Ok button to automatically enter the numerals 1,2,3,4,5,6 (in that order, as the elements of the set G), as row and column headings. These will also appear as the entries in the first row and first column of the table as shown in Figure G-1. • Enter the elements of the subset {1,2,3} as shown in Figure G-2 so that they form a subsystem. • Finally, enter the other numerals until the whole table is completely filled up as shown in Figure G-3 and G-4. Each entry must satisfy the chosen Axioms in order to complete the desired table.

  13. SETTINGS OF THE PARAMETERS DIALOG BOX Figure G-P. This box shows the settings for the system (G,)to be constructed. Note that the element 1 has been chosen as the identity element and that A1, A3, A4, and A5 have been activated.

  14. Characteristic Patterns In filling out the Cayley table S(G), the pattern (or position) of entries is determined by the Axioms the system satisfies. These are automatically satisfied if they are activated in the Parameters dialog box under the SEMI-AUTO MODE. • P(A1) Closure Pattern: All entries gij = gi gj of S(G) are elements of the set G and there is exactly one entry at each intersection of a given row i and a given column j of S(G). • P(A2) Identity Pattern: In S(G), the entries in row e are identical to the entries in column e, that is, gek = gk = gke for all k = 1,…,n, where ge is the identity element of G. • P(A3) Inverse Pattern: The identity element ge  G appears exactly once in each row and in each column of S(G) such that gab = ge = gba . • P(A4) Unique Solution Pattern: Each element gb G appears exactly once in each row x and in each column y of S(G), that is, gxa = gb = gka iff k = x and gay = gb = gak iff k = y. • P(A5) Commutative Pattern: The entries of S(G) are such that gab = gba for all a,b = 1,…,n; S(G) is symmetric.

  15. CAYLEY TABLE OF THE SYSTEM (G,) UNDER CONSTRUCTION Figure G-1. This table shows the entries in the 1st row and 1st column, where the element 1 is the chosen identity.

  16. PARTIAL ENTRIES OF THE CAYLEY TABLE OF (G,)UNDER CONSTRUCTION Figure G-2. This table shows the entries for the required subsystem ({1,2,3},) of order m = 3.

  17. PARTIAL ENTRIES OF THE CAYLEY TABLE OF (G,)UNDER CONSTRUCTION Figure G-3. This table shows additional entries for the elements of the subset {1,2,3} that satisfy A4.

  18. COMPLETED CAYLEY TABLE OF THE SYSTEM (G,) Figure G-4. This table shows the entries for the elements of the subset {4,5,6} that satisfy A4. This completes the construction of the Cayley table of the system (G,).

  19. HOW TO ANALYZE A SYSTEM • To analyze the constructed system (G,), open the Analyze menu and click on Axioms Test. • Now click on Special properties test to determine its weak associative properties (if any)like power-associative, semi automorphic inverse properties, etc.Then click on Structural properties test and choose Subsystems test, etc.

  20. HOW TO DISPLAY RESULTS OF ANALYSIS • To display the results of analyses done on (G,), go to the Display menu and choose System properties of(G,), next Special properties of(G,), then Subsystems of(G,), etc.

  21. DISPLAY: SUMMARY OF ANALYSIS OF (G,) This display simply shows the summary of the analysis done on the system (G,). It also shows that (G,) is an Abelian NAFIL loop of order 6.

  22. DISPLAY: SPECIAL PROPERTIES OF (G,) The system (G,) was found to have the following special properties: AIP, AAIP, SAIP, and FL. The first three properties are related.

  23. DISPLAY: SUBSYSTEMS OF (G,) This display shows that (G,) has three subsystems all of which are abelian groups. In particular, it shows that the subset {1,2,3} forms a subsystem.

  24. Other FINITAS Functions FINITAS can also do the following: • Determine the powers (Left/Right), orders (Left/Right) of elements, as well as generators (if any) of a loop. • Determine (if any) the normal subsystems of a loop and their factor systems. • Determine (if any) the nuclei (left, middle, right), nucleus, and center of a loop. • Determine certain associated matrices of the elements of a system and their permutation representations. • Compare two systems for isomorphism, homomorphism, and isotopy. • Import the text file of a Cayley table and convert it into a valid FINITAS table file.

  25. FINITAS - Presentations FINITAS has been presented in the following international conferences: • 2000 - International Congress on Ghiyath Al-Din Jamshid Kashani, Kashan University, Iran • 1997 - 24th IBC/ABI International Congress on Arts and Communications, Keble College, Oxford University, England • 1996 - 2nd European Congress of Mathematics Janos Bolyai Mathematical Society, Budapest, Hungary • 1995 - 2nd Asian Mathematics Conference, Suranaree University of Technology, Nakhon Ratchasima, Thailand.

  26. ACKNOWLEDGEMENT FINITAS was developed at the PUP SciTech R&D Center with the support of the National Research Council of the Philippines (NRCP) and the Philippine Council for Advanced Science and Technology Research and Development (PCASTRD) of the Department of Science and Technology (DOST). I also wish to acknowledge the invaluable help of my Research Assistants Alexander S. Carrascal, Renilda S. Layno, Aurea Z. Rosal, my able programmers Allan Dimanlig and Nero B. Leona, and the other members of my research staff.

  27. ANNOUNCEMENT FINITAS is available for trial purposes to interested delegates of the LOOPS’03 International Conference. An End User License Agreement (EULA) is available. • The FINITAS V1.1 files can be copied and the User’s Manual can be reproduced, as well as several papers related to FINITAS and its applications. For inquiries, send an e-mail to: Raoul E. Cawagas raoulec@pacific.net.ph The

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