Deanship of Scientific Research

DEANSHIP OF SCIENTIFIC RESEARCH

KFUPM RESEARCH FUNDED PROJECT WITH ONE SEMESTER RELEASE TIME NO. IP-2007/15

Title of Proposal: Semigroups of Order-decreasing Transformations on Posets

Duration of Project (in months): 18 Proposed Starting Date: July, 2007 Ending Date: December, 2008 Total Project Cost (SR): 43,600 Submitted by: Dr. Abdullahi Umar, Associate Professor, Math. Sciences

Date: March 11, 2007.

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APPROVALS:

Chairman:………………………………………………………………..…………………… Date: ……………………

Department: Mathematical Sciences

Chairman, Research Committee: …………………………………………… Date: ……...…………

Vice Rector for Graduate Studies and Scientific Research: ……………………………….

Date: …………..………

RESEARCH PROJECT NO.: IP-2007/15

Title of Proposal: Semigroups of Order-decreasing Transformations on Posets

Submitted by: Dr. Abdullahi Umar, Associate Professor, Math. Sciences 3

TABLE OF CONTENTS

Abstract (English) ……………………………………………………………………………. 4

Introduction …………………………………………………………………………………… 5

Literature Review ……………………………………………………………………………. 5

Objectives and Description of Proposed Research ………….…………………………….. 6

Scheduling of the Proposed Research …………………………………………………. 6

Methodology …………………………………………………. 6

Personnel Requirements …………………………………………………………………....6

Monitoring and Evaluation …………………………………………………………………....6

Utilization Plan ……………………………………………………………………………. 7

References …………………………………………………………………………….………. 7

Budget…………………………………………………………………………………………….. 9

Suggested Reviewers …………………………………………………………………....10

Curriculum Vitae …………………………………………………………………………… 11 4

1. Abstract

Let X be a partially ordered set (poset) and consider the semigroups of order-decreasing

(increasing) full, partial and partial one-one transformations of X: DX, PDX and IDX, respectively. For various types of posets we would like to investigate the algebraic, rank and combinatorial properties of these semigroups. 5

2. Introduction The project proposes to initiate a general study of the semigroups of order-decreasing (increasing) transformations on partially ordered sets (posets). Hitherto, these semigroups have been extensively studied in the case of totally ordered sets, but very few results were obtained in the case of posets. One of the main difficulties in this work is while there is only one totally ordered set of size n up to isomorphism, there are many non-isomorphic posets of the same size. In fact, for size 2, there are 2 posets; for size 3 there are 5 posets, for size 4 there are 11 posets: and their number increase geometrically (if not exponentially) with increase in n. In particular, we propose to find answers to the following questions: for which posets are these semigroups abundant/idempotent-generated; count the number of elements in certain natural equivalence classes; find their minimal generating sets.

The primary aim of the project is to obtain usable characterizations of the poset X, for which D(X) (the semigroup of all order-decreasing full transformations of X) is abundant/idempotent- generated; settle some natural combinatorial questions and find minimal generating systems.

3. Literature Review The theory of semigroups started when the theories of groups and rings were well established fields exerting their influence of domain on the whole of algebra and mathematics in general. Nevertheless the theory has come along way since then. Its scope has widened remarkably to embrace many aspects of computer science. The study of semigroups of transformations of various kinds is central to (algebraic and topological) semigroup theory. Arguably, one of the fore-runners in research in this area is my thesis advisor, Professor John M. Howie (the now retired Regius Professor of mathematics at St Andrews but still an active mathematician). In my Ph. D. thesis (1992), for a totally ordered set X, I initiated a general study of the semigroups of order-decreasing (increasing) transformations (full, partial and partial one-one) and subsequently many results pertaining to these semigroups were published by Higgins (1993, 1997), Umar (1992, 1993, 1994, 1996a,b; 1997, 1999, 2003), Howie (1994), and Saito, Aoki and Kajitori (1996). Earlier, Pin (1984) and recently Laradji and Umar (2004a,b) both contain results associated with some of these semigroups. Most of these results were themselves analogues of earlier results by Howie (1966, 1971, 1978), Lyapin (1974), Gomes and Howie (1987, 1992), Borwein et. al. (1989), Howie and McFadden (1990) and Garba (1990,1994a, 1994b) on various semigroups of transformations. 6

By contrast with the earlier semigroups studied, the semigroups of order-decreasing transformations of a totally ordered set fail to be regular in general, but they turn out to be abundant semigroups (these are one the most successful generalizations of regular semigroups introduced by Fountain (1982)). This gave rise to a class of natural examples of abundant semigroups.

4. Objectives and description of the Proposed Research We shall begin by characterizing the Green's relations and their starred analogues. This will help in understanding the algebraic structure and properties of these classes of semigroups and to the abstract classes of semigroups to which they belong. Next we tackle combinatorial questions by counting all elements and various kind of special elements in these semigroups - these will lead naturally to determining minimal generating sets each of these semigroups. We intend to pursue the following tasks: Task 1. Investigate the algebraic properties of these semigroups. Task 2. Investigate the combinatorial properties of these semigroups. Task 3. Investigate the rank properties of these semigroups.

5. Methodology To investigate the algebraic properties we first study the Green's relations and their starred analogues which usually lead to deeper understanding of semigroups. To investigate the combinatorial properties of these semigroups to utilize the approach we successfully used in Laradji and Umar [12 - 17]. To investigate the rank properties we use whichever of the techniques of Gomes and Howie [7], Garba [1990, 1994b] and Umar [18], to be appropriate.

6. Scheduling of the Proposed Research Time Table Task Months Task 1 8 Task 2 6 Task 3 4 7

This proposal is to be carried out at KFUPM and Wilfrid Laurier University, Ontario according to the following schedule.

Period Month 1 - 6 Month 7-12 Month 13 - 18 Institution KFUPM Wilfrid KFUPM Laurier

7. Personnel Requirement I am soliciting a one semester release time to be spent at the department of mathematics, Wilfrid Laurier University, Ontario, Canada, where I will be the guest of Professor Sydman Bulman-Fleming who is an accomplished semigroup theorist.

8. Monitoring and Evaluation I expect my results to be published in international journals.

9. Utilization Plan I expect my results to yield a very rich source of natural examples for some 'natural' classes of (abstract) semigroups, which will increase our understanding of them, and may even lead to structure theorems.

10. References

Borwein, D., Rankin, S., and Renner, L. Enumeration of injective partial transformations. Discrete Math. (1989), 73: 291–296. Fountain, J. B. Abundant semigroups. Proc. London Math. Soc. (1982), 44: 103–129. Garba, G. U. Idempotents in partial transformation semigroups. Proc. Roy. Soc. Edinburgh (1990), 116A: 359–366. Garba, G. U. On the nilpotent rank of partial transformation semigroups. Portugaliae Mathematica (1994a), 51: 163–172. Garba, G. U. On the nilpotent ranks of certain semigroups of transformations. Glasgow Math. J. (1994b), 36: 1–9. Gomes, G. M. S., and Howie, J. M. Nilpotents in finite symmetric inverse semigroups. Proc. Edinburgh Math. Soc. (1987), 30: 383–395. Gomes, G. M. S., and Howie, J. M. On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum (1992), 45: 272–282. Higgins, Peter M. Combinatorial results for semigroups of order-preserving mappings. Math. Proc. Camb. Phil. Soc. (1993), 113: 281–296. 8

Higgins, Peter M. A proof of Simon's theorem on piecewise testable languages. Theoret. Comput Sci. (1997), 178: 257–264. Howie, J. M. The subsemigroup generated by the idempotents of a full transformation semigroups. J. London Math. Soc. (1966), 41: 707–716. Howie, J. M. Products of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. (1971), 17: 223–236. Howie, J. M. Idempotents generators in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Math. (1978), 81: 317–323. Howie, J. M. Combinatorial and probabilistic results in transformation semigroups. Words, languages and combinatorics, II (Kyoto, 1992), 200--206, World Sci. Publ., River Edge, NJ, (1994). Howie, J. M., and McFadden, R. B. Idempotent rank in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh (1990), 114: 161–167. Laradji, A. and Umar, A. On certain finite semigroups of order-decreasing transformations I, Semigroup Forum (2004a), 69: 184–200. Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. (2004b), 7: 04.3.8, 14 pp. Lyapin, E. S. "Semigroups," Translations of mathematical monographs, Vol. 3. American Math. soc. Providence, R. I. (1974). Pin, J. E. Varieties of formal languages Masson, Paris, 1984. English translation, translated by A. Howie (North Oxford Academic Publishers Ltd., 1986). Saito, T. Aoki, K. Kajitori, K. Remarks on isomorphisms of regressive transformation semigroups. Semigroup Forum (1996), 53: 129–134. Umar, A. Semigroups of order-dereasing transformation, Ph. D Thesis, University of St Andrews (1992). Umar, A. On the semigroups of order-decreasing finite full transformations, Proc. Roy. Soc. Edinburgh (1992), 120A: 129–142. Umar, A. On the semigroups of partial one-to-one order-decreasing finite transformations, Proc. Roy. Soc. Edinburgh (1993), 123A: 355–363. Umar, A. On the ranks of certain finite semigroups of order-decreasing transformations. Portugaliae Mathematica Vol. 53 Fasc. 1 (1996a), 23-34. Umar, A. Semigroups of order-decreasing transformations: the isomorphism theorem. Semigroup Forum Vol. 53 (1996b), 220-224. Umar, A. On certain infinite semigroups of order-decreasing transformations I. Communications in Algebra 25 (9) (September,1997), 2987-2999 Umar, A. A class of (0)-idempotent-free transformation semigroups. Semigroup Forum 59 (1999), 74-78. Umar A., On certain infinite semigroups of order-increasing transformations II. Arabian Journal for Science and Engineering. Vol. 28, 2A (2003), 203-210 9

11. Budget Estimated Costs i) Trip to Wilfrid Laurier University, Ontario 10000 SR ii) Conference attendance 10000 SR iii) Compensation 21600 SR iv) Miscellaneous 1000 SR v) Secretary 1000 SR

Total 43600 SR 10

12. Suggested Reviewers

1. Professor K. D. Magill, Jr. Department of Mathematics, University at Buffalo The New York State Univ. Buffalo NY 14260-2900. [email protected]

2. Professor P. M. Higgins Department of Mathematics, University at Buffalo The New York State Univ. Buffalo NY 14260-2900. [email protected]

3. Professor J. B. Fountain Department of Mathematics, University of York, Heslington, York YO1 5DD England. [email protected]

4. Professor R. B. McFadden Department of Mathematics, Gardiner Hall, 2nd Floor, Belknap Campus University of Louisville, Louisville KY 40292. [email protected]

5. Professor Gracinda M. S. Gomes Centro de Algebra Universidade de Lisboa Av. Prof. Gama Pinto, 2 1699 Lisboa Codex Portugal. [email protected]

6. Professor Nik Ruskuc School of Mathematics, University of St Andrews, St Andrews KY16 9SS, Scotland. [email protected] 11

CURRICULUM VITAE Personal Data

Name: Abdullahi Umar Date of Birth: Wednesday February, 21st 1962 – Hadejia Marital Status: Married – four children Address: Department of Mathematical Sciences King Fahd University of Petroleum & Minerals P. O. Box 672 Dhahran 31261, Saudi Arabia

Phone: 00966-3-8602720 (Office) 00966-3-8605065 (Res.)

Fax: 00966-3-8602340 E-mail: aumar@ kfupm.edu.sa Present Position: Associate Professor Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran 31261, SAUDI ARABIA

Education

1978 Government College Kano (Now Rumfa College), Nigeria 1983 B.Sc., (Hons.), (First Class) Mathematics, Ahmadu Bello University (ABU), Zaria, Nigeria. 1986 M.Sc., Mathematics, Ahmadu Bello University (ABU), Zaria, Nigeria. 1992 Ph.D., Mathematics, Algebra, St Andrews University, Scotland, U.K. Major: Mathematics Area of Specialization: Algebraic semigroups: transformation semigroups 12

Career History and Teaching Experience

(a) Honorary Graduate Assistant, School of Basic Studies, Zaria (1983):

O' Level Math. for non-math. students

(b) Graduate Assistant, ABU, Zaria (1984-1986):

D102 Algebra I M101 Mathematical Methods I D101 Calculus I D104 Vectors

(c) Assistant Lecturer, ABU Zaria (1986-1989):

D103 Trigonometry D202 Algebra II M101 Mathematical Methods I M109 Maths for Pharmacy D202 Algebra II M101 Mathematical Methods I M108 Supplementary Maths. M201 Mathematical Methods II M209 Mathematical Methods for Teachers

(d) Tutorial Assistant, University of St Andrews (1989-1992):

MT2002 Algebra and Analysis MT2005 Combinatorics

(e) Lecturer I, ABU, Zaria (1993-1995):

MATH101 Calculus MATH205 Introduction to Abstract Algebra MATH102 Algebra MATH 206 Abstract algebra I MATH 505 Semigroups I MATH 506 Semigroups II 13

(f) Senior Lecturer, University of Abuja, Nigeria (1995-1997):

MTH101 Algebra MTH205 Elementary Differential Equations MTH403 Measure and Integration MTH106 Calculiu MTH310 Discrete Math. MTH404 Fluid Dynamics MTH410 Theory of Rings and Fields CSC602 Linear Algebra MTH604 Theory of Computations

(g) Assistant Professor, KFUPM (Oct. 1997- Apr. 2005):

MATH131 Finite Mathematicsc, c MATH201 Calculus III MATH232 Introduction to sets and Structures MATH102 Calculus II MATH202 Elements of Differential Equationsc MATH 002 Preparatory Mathematics II MATH280 Introduction to Linear Algebra MATH101 Calculus I MATH012 Finite Mathematics for Diploma MATH132 Applied Calculusc c served as coordinator

(h) Associate Professor, KFUPM (2005-Present):

MATH131 Finite Mathematicsc, c MATH596 Reading and Research II MATH202 Elements of Differential Equationsc MATH132 Applied Calculusc c served as coordinator

Supervised Senior Projects

Akweel, Ali. “The Bicyclic semigroup”. Math 490 (Seminar in Mathematics, May 2001). Al-Shammari, Sutam A. “Euclidean Domains”. Math 490 (Seminar in Mathematics, May 2003) Katebah, Ayman. “Transformation Monoids and an Automaton”. Math 490 (Seminar in Mathematics, December 2005) 14

Supervised M.Sc. Thesis

Omar, Syed. “Some properties of Bicyclic Extensions”. M. S. Thesis, Department of Mathematical Sciences KFUPM, (December 1998).

Academic Advisor I have been an academic advisor to undergraduate maths major students since 2002, and of all undergraduate maths. major students since 2004.

Course(s) Development:

 Developed a course in Discrete maths. (with Drs. Abu-Sbeih and Laradji).

Administrative Experience

 Acting Head of Mathematics at University of Abuja (June, 1997).  Chairman, Algebra Day 2005 Organizing Committee (2004/05).  Coordinator, Algebra and Number Theory Scientific Group (2004/05)

Prizes and Awards

1. AHMADU COOMASSIE PRIZE for the best graduating student in mathematics at Ahmadu Bello University, Zaria, Nigeria (1983). 2. Federal Government of Nigeria special (Ph. D.) scholarship (1989-92). 3. Carnegie Trust for the Universities of Scotland Research Award at St Andrews University, Scotland (1996) 4. Royal Society Visiting Research Fellowship Award at the University of Essex (1999).

Research

Research interests

My current research interests include transformation semigroups of various types, especially their combinatorial and rank properties. I also have interest in discrete mathematics and recreational mathematics which include topics like magic squares and cubes, calendars and puzzles.

Refereed Journal Publications:

1. Umar, A. On the semigroups of order-decreasing finite full transformations. Proc. Roy. Soc. Edinb. Sect. A 120 (1992), 129-142. 2. Umar, A. On the semigroups of partial one-to-one order-decreasing finite transformations. Proc. Roy. Soc. Edinb. Sect. A 123 (1993), 355-363. 3. Umar, A. On the ranks of certain finite semigroups of order-decreasing transformations. Portugaliae Mathematica Vol. 53 Fasc. 1 (1996), 23-34. 15

4. Umar, A. A class of quasi-adequate transformation semigroups. Portugaliae Mathematica Vol. 51 Fasc. 4 (1994), 553-570. 5. Umar, A. Semigroups of order-decreasing transformations: the isomorphism theorem. Semigroup Forum Vol. 53 (1996), 220-224. 6. Makanjuola, S. O. and Umar, A. On a certain subsemigroup of the bicyclic semigroup. Communications in Algebra 25 (2) (February, 1997), 509-519. 7. Umar, A. Enumeration of certain finite semigroups of transformations. Discrete Math. 189 (1998), 291-297. 8. Umar, A. On certain infinite semigroups of order-decreasing transformations I. Communications in Algebra 25 (9) (September,1997), 2987-2999. 9. Umar, A. Some remarks about Fibonacci groups and semigroups. Communications in Algebra 25 (12) (December, 1997), 3973-3977. 10. Umar, A. A class of (0)-idempotent-free transformation semigroups. Semigroup Forum 59 (1999), 74-78. 11. Umar A., On certain infinite semigroups of order-increasing transformations II. Arabian Journal for Science and Engineering. Vol. 28, 2A (2003), 203-210. 12. Laradji, A. and Umar, A., “Combinatorial results for semigroups of order-preserving partial transformations”. Journal of Algebra 278 (2004), 342-359. 13. Laradji, A. and Umar, A., “On certain finite semigroups of order-decreasing transformations I”. Semigroup Forum 69 (2004), 184-200. 14. Laradji, A. and Umar, A., “On the number of nilpotents in the partial symmetric semigroup”. Communications in Algebra, 32 (2004), 3017-3023. 15. Laradji, A. and Umar, A., “Combinatorial results for semigroups of order-decreasing partial transformations”. J. Integer Sequences 7 (2004), 04.3.8. 16. Laradji, A. and Umar, A., “Asymptotic results for semigroups of order-preserving partial transformations”. Communications in Algebra 34 (2006), 1071-1075. 17. Laradji, A. and Umar, A., “Combinatorial results for semigroups of order-preserving full transformations”. Semigroup Forum 72 (2006), 51-62. 18. A. Umar , B. Yushau and B. M. Ghandi, “Convolution of two series” Australian Senior Maths Journal. (accepted)

 Articles number [1, 2, 3 & 8] are from my Ph.D. thesis. Article [5] is a slight extension of a major result in my Ph. D. thesis, but more importantly the main proof is more elegant than the one given in the thesis.  In almost all mathematical journals, authors' names are listed alphabetically (by last names) and not in order of contribution as in other scientific journals.

Conference Papers:

19. Umar, A. “Semigroups of order-decreasing full transformations.” Proceedings of the international conference, Colchester, UK, August 3--6, 1993. Colchester: University of Essex, Dept. of mathematics, August (1994), 94-98. 20. Umar, A. “On a class of orthodox transformation semigroups.” 15th Annual Conference of the Nigerian Mathematical Society, University of Nigeria, Nsukka, April 1994. 21. Umar, A. “On some generalizations of the hat problem and transformation semigroups.” International Algebraic Conference, Ekaterinburg, Russia, August 29- Sept. 3, 2005, p. 203. 22. A. Umar, B. Yushau and B. M. Ghandi, (2006)“Patterns in convolution of two series”, in Stewart, S. M., Olearski, J. E. and Thompson, D. (Eds), Proceedings of the 16

Second Annual Conference for Middle East Teachers of Science, Mathematics and Computing (pp. 95-101). METSMaC: Abu Dhabi.

Technical Reports:

23. Umar, A., “Construction of Even Order Magic Squares,” Technical Report No. 233, (July 1998) Department of Mathematical Sciences, KFUPM. 24. Higgins, P. M. and Umar, A., “Semigroups of weak V-Stabilizer mappings,” Technical Report No. 238, (November 1998) Department of Mathematical Sciences, KFUPM. 25. Higgins, P. M. and Umar, A., “Semigroups of Order-decreasing Transformations: Some Fundamental Congruences,” Technical Report No. 268, (September 2001) Department of Mathematical Sciences, KFUPM. 26. Laradji, A. and Umar, A., “On certain finite semigroups of order-decreasing transformations I,” Technical Report No. 298, (May 2003) Department of Mathematical Sciences, KFUPM. (Appeared as Journal Publication #13.) 27. Laradji, A. and Umar, A., “On the number of nilpotents in partial symmetric semigroup,” Technical Report No. 305, (June 2003) Department of Mathematical Sciences, KFUPM. (Appeared as Journal Publication #14.) 28. Laradji, A. and Umar, A., “Combinatorial results for semigroups of order-decreasing partial transformations,” Technical Report No. 306, (June 2003) No. 305, (June 2003) Department of Mathematical Sciences, KFUPM. (Appeared as Journal Publication #15.) 29. Laradji, A. and Umar, A., “Asymptotic results for semigroups of order-preserving partial transformations,” Technical Report No. 310, (October 2003) Department of Mathematical Sciences, KFUPM. 30. Laradji, A. and Umar, A., “On the number of decreasing and order-preserving partial transformations,” Technical Report No. 312, (January 2004) Department of Mathematical Sciences, KFUPM. 31. Laradji, A. and Umar, A., “Combinatorial results for semigroups of order-preserving partial transformations,” Technical Report No. 313, (January 2004) Department of Mathematical Sciences, KFUPM. (Appeared as Journal Publication #16.) 32. Laradji, A. and Umar, A., “On some generalizations of the hat problem and transformation semigroups,” Technical Report No. 358, (October 2006) Department of Mathematical Sciences, KFUPM.

Submitted for publication:

33. Laradji, A. and Umar, A., “Combinatorial results for the symmetric inverse semigroup”. 34. Laradji, A. and Umar, A., “Lattice paths and partial order-preserving transformations”.

Work In Progress: 35. Combinatorial Properties of various classes of transformation Semigroups (with A. Laradji). 17

Refereeing and Citation 

Article number Ph.D. Paper # # # Thesis #1 2 5 6 #8 Number of 2 9 1 3 1 1 citations

Refereed 2 papers for Communications in Algebra by (1) ARAUJO, J. AND KONIECZNY, J. ENTITLED: SEMIGROUPS OF TRANSFORMATIONS PRESERVING AN EQUIVALENCE AND A CROSS- SECTION; (2) XIULIANG, Y. ENTITLED: H-TRIVIAL SEMIBANDS AND DIGRAPHS.

Refereed 1 paper for Algebra Colloquium by (1) KUYUCU AND VATANSEVER ENTITLED: ON SUBSEMIGROUPS OF STONE- CECH COMPACTIFICATION FOR CERTAIN SEMIGROUPS.

Recent Conferences and Seminars

1. Workshop entitled: Introduction to WebCT, Academic Development Center, KFUPM, July, 2003. 2. Workshop entitled: Instructional Design for Online Courses, Academic Development Center, KFUPM, January, 2004. 3. ACM-SIAM Symposium on Discrete Algorithms, Vancouver, 23-25 January, 2005. 4. International Algebraic Conference, Ekaterinburg, Russia, Aug. 29-Sept. 3, 2005. 5. Second Annual Conference for Middle East Teachers of Science, Mathematics and Computing (METSMaC), Abu Dhabi, UAE, 14-16 March 2006. 6. What is the Next Term of a Sequence?. UAE Math Day, University of Sharjah, UAE, 27 April 2006. 18

Recent Lectures/Talks Delivered:

1. Monoids, Automata and Languages, KFUPM 2001. 2. Combinatorial Properties of Transformation Semigroups: An Update, KFUPM 2003. 3. On a generalization of the hat problem, KFUPM 2004.. 4. On some generalizations of the hat problem, Bayero University, Kano, Nigeria, June 2004. 5. On some generalizations of the hat problem, A. B. U., Zaria, Nigeria, June 2004. 6. Poster Presentation: Patterns in Convolutions of two Series. METSMaC, Abu Dhabi, UAE, 14-16 March 2006. 7. Predicting the Next Term of a Sequence. KFUPM 2006. 8 What is the Next Term of a Sequence?. UAE Math Day, University of Sharjah, UAE, 27 April 2006.

Professional Visits

Visited Essex University for 6 weeks from Sept. – Oct. 1993. Visited St Andrews University for 4 months from Oct. – Feb. 1996/97. Visited Essex University for 6 weeks from Feb. – Mar. 1997. Visited Essex University for 6 months from July – Dec. 1999. Visited Ahmadu Bello University, Zaria, Nigeria for 7 weeks from June – Aug. 2006

Participation in Committees

Departmental Committees:

Standing Committees: 1. Library Committee (97-98) 2. Algebra and Number Theory Group (97- Present) 3. Graduate -MS program Committee (98-99) 4. Undergraduate Committee (00-01, & 02-06) 5. Search Committee (01-02 & 02-03)

Ad-Hoc Committees: 1. Distinguished Teacher and Advisor Award (00-01, 01-02 & 02-03) 2. Text book Review Committee – MATH 232/MATH 305 (2004), 3. Algebra and Applications Day 2005, 1st March 2005, Dept. of Math. Sciences, KFUPM. Chairman Organizing Committee 4. Algebra Workshop, 19 – 21 March 2006, Dept. of Math. Sciences, KFUPM. Member Organizing Committee 19

CHECK LIST

Please indicate (x) if present. (x) 1. Title Page a) English (x) b) Arabic ( ) (x) 2. Table of Contents (x) 3. Abstract a) English (x) b) Arabic ( ) (x) 4. Introduction (x) 5. Literature Review (x) 6. Project Objectives a) Basic (x) b) Applied ( ) (x) 7. Description of the Proposed Research (N/A) 8. Experimental Design and Procedure (x) 9. Scheduling of the Proposed Research (x) 10. Requirements (x) 11. Monitoring and Evaluation (x) 12. Utilization Plan (x) 13. References (x) 14. Budget (x) 15. Suggested Reviewers (x) 16. CV

Signature:_

Name of the Principal Investigator: Dr. Abdullahi Umar

Date:__March. 11, 2007____