ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper Characterization and axiomatization of all semigroups whose square is group Characterization and axiomatization of all semigroups whose square is group Hooshmand M.H. Shiraz Branch, Islamic Azad University 01 11 2015 2 2 1 8 04 08 2015 26 04 2016 Copyright © 2015, Yazd University. 2015 http://as.yazd.ac.ir/article_741.html

In this paper we give a characterization for all semigroups whose square is a group. Moreover, we axiomatize such semigroups and study some relations between the class of these semigroups and Grouplikes,introduced by the author. Also, we observe that this paper characterizes and axiomatizes a class of Homogroups (semigroups containing an ideal subgroup).  Finally, several equivalent conditions for a semigroup \$S\$ with \$S^2leq S\$ (the square-group property) will be considered.

Ideal subgroup grouplike homogroup class united grouplike real grouplike
 A. H. Cli ord and D. D. Miller, Semigroups having zeroid elements, Amer. J. Math. vol. 70 (1948), 117-125.  D. P. Dawson, Semigroups Having Left of Right Zeroid Elements, Acta Scientiarum Mathematicarum, XXVII (1966), 93-957.  M.H.Hooshmand, Grouplikes, Bull. Iran Math. Soc., Bull. Iran. Math. Soc., vol. 39, no. 1 (2013), 65-86.  M.H.Hooshmand and H. Kamarul Haili, Decomposer and Associative Functional Equations, Indag. Mathem., N.S., vol.18, no. 4 (2007), 539-554.  M.H.Hooshmand, Upper and Lower Periodic Subsets of Semigroups, Algebra Colloquium, vol. 18, no.3 (2011), 447-460.  M.H. Hooshmand and H. Kamarul Haili, Some Algebraic Properties of b-Parts of Real Numbers,  Siauliai Math.Semin., vol.3, no.11 (2008), 115-121.  M.H. Hooshmand and S. Rahimian, A study of regular grouplikes, J. Math. Ext., vol. 7, no. 4 (2013), 1{9.  R. P. Hunter, On the structure of homogroups with applications to the theory of compact connected semigroups, Fund. Math. vol. 52 (1963), 69-102.  A. Nagy, Special Classes of Semigroups, Kluwer Academic Publishers, 2001.  G. Thierrin, Contribution a la theorie des equivalences dans les demi-groupes, Bull. Soc. Math. France, vol. 83 (1955), 103-159.
ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex? VISWESWARAN S. Saurashtra University, Rajkot, India PARMAR A. Saurashtra University, Rajkot, India 01 11 2015 2 2 9 22 09 04 2016 31 05 2016 Copyright © 2015, Yazd University. 2015 http://as.yazd.ac.ir/article_765.html

The rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. Let \$R\$ be a ring. Let \$mathbb{A}(R)\$ denote the set of all annihilating ideals of \$R\$ and let \$mathbb{A}(R)^{*} = mathbb{A}(R)backslash {(0)}\$. The annihilating-ideal graph of \$R\$, denoted by \$mathbb{AG}(R)\$  is an undirected simple graph whose vertex set is \$mathbb{A}(R)^{*}\$ and distinct vertices \$I, J\$ are joined by an edge in this graph if and only if \$IJ = (0)\$. The aim of this article is to classify rings  \$R\$ such that \$(mathbb{AG}(R))^{c}\$ ( that is,  the complement of \$mathbb{AG}(R)\$)   is connected and admits a cut vertex.

N-prime of \$(0)\$ B-prime of \$(0)\$ complement of the annihilating-ideal graph of a commutative ring vertex cut and cut vertex of a connected graph
 G Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M.J. Nikmehr, and F. Shaiveisi, The classi cation of annihilating-ideal graph of commutative rings, Alg. Colloquium, 21, 249 (2014), doi:10.1143/S1005386714000200.  G. Aalipour, S. Akbari, R. Nikandish, M.J. Nikmehr, and F. Shaiveisi, On the coloring of the annihilating- ideal graph of a commutative ring, Discrete Math., 312 (2012), 2620-2625.  .D.F. Anderson, M.C. Axtell, J.A. Stickles Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian perspectives, M. Fontana, S.E. Kabbaj, B. Olberding, and I. Swanson (Editors), Springer-Verlag, New York, 2011, 23-45.  D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Alg. 217 (1999),  M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mas- sachusetts, 1969.  M.C. Axtell, N. Baeth, and J.A. Stickles, Cut vertices in zero-divisor graphs of nite commutative rings, Comm. Alg., 39(6) (2011), 2179-2188.  R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New york,  I. . Beck, Coloring of commutative rings, J. Alg. 116 (1988), 208-226.  M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Alg. Appl. 10 (2011),  M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Alg. Appl.10 (2011),  B. Cotee, C. Ewing, M. Huhn, C.M. Plaut, and E.D. Weber, Cut-Sets in zero-divisor graphs of nite commutative rings, Comm. Alg. 39(8) (2011), 2849-2861.  R. Gilmer and W. Heinzer, The Laskerian property, power series rings and Noetherian spectra, Proc. Amer. Math. Soc. 79(1) (1980), 13-16.  W. Heinzer and J. Ohm, Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158(2) (1971),  W. Heinzer and J. Ohm, On the Noetherian-like rings of E.G. Evans, Proc. Amer. Math. Soc. 34(1) (1972),  M. Hadian, Unit action and geometric zero-divisor ideal graph, Comm. Alg. 40 (2012), 2920-2930.  I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.  T. Tamizh Chelvam and K. Selvakumar, On the connectivity of the annihilating-ideal graphs, Discuss. Math. Gen. Alg. Appl. 35 (2015), 195-204.  S. Visweswaran, Some results on the complement of the zero-divisor graph of a commutative ring, J. Alg. Appl. 10(3) (2011), 573-595.  S. Visweswaran, Some properties of the complement of the zero-divisor graph of a commutative ring, ISRN Alg. 2011 (2011), Article ID 591041, 24 pages.  S. Visweswaran and Hiren D. Patel, Some results on the complement of the annihilating ideal graph of a commutative ring, J. Algebra Appl. 14 (2015), doi: 10.1142/S0219498815500991, 23 pages.  S. Visweswaran, When does the complement of the zero-divisor graph of a commutative ring admit a cut vertex?, Palestine J. Math. 1(2) (2012), 138-147.
ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper Ultra and Involution Ideals in \$BCK\$-algebras Ultra and Involution Ideals in \$BCK\$-algebras Saidi Goraghani Simin Farhangian University Borzooei R. A. Shahid Beheshti University 01 11 2015 2 2 23 36 24 07 2016 24 07 2016 Copyright © 2015, Yazd University. 2015 http://as.yazd.ac.ir/article_784.html

In this paper, we define the notions of ultra and involution ideals in \$BCK\$-algebras. Then we get the relation among them and other ideals as (positive) implicative, associative, commutative and prime ideals. Specially, we show that in a bounded implicative \$BCK\$-algebra, any involution ideal is a positive implicative ideal and in a bounded positive implicative lower \$BCK\$-semilattice, the notions of prime ideals and ultra ideals are coincide.

\$BCK\$-algebra (associative commutative positive implicative implicative) ideal ultra ideal involution ideal
 R. A. Borzooei, J. Shohani, Fraction structures on bounded implicative BCK-algebras, Word Academy of Science, Engineering and Thechnology, 49, 1084-1090 (2009).  O. Heubo-Kwegna and J. B. Nganou, A Global Local Principle for BCK-modules, International Journal of Algebra, 5(14), 691-702 (2011).  Y. Huang, BCI-algebra, Science Press, Beijing (2006).  Y. Imai and K. Iseki, On axiom systems of propositional calculi, Proceedings of the Japan Academy, 42, 19-21 (1966).  K. Iseki, On ideals in BCK-algebras, Mathematics Seminar Notes, 3, 1-12 (1975).  K. Iseki and S. Tanaka, Ideal theory of BCK-algebras, Mathematica Japonica, 21, 351-366 (1976).  P. Jiayin, Normed BCK-algebras, Advances in Mathematics, 4, 492-500 (2011).  J. Meng and Y. B. Jun, BCK-algebras, Kyung Moon Sa Co, seoul, (1994).  Z. M. Samaei and M. A. N. Azadani, A Class of BCK-algebras, International Journal of Algebra, 28, 1379-1385 (2011).
ASTA Yazd University Algebraic Structures and Their Applications 2382-9761 Yazd University 10 Research Paper The structure of a pair of nilpotent Lie algebras The structure of a pair of nilpotent Lie algebras Arabyani Homayoon Islamic Azad University Fadravi Hadi Hosseini Islamic Azad University 01 11 2015 2 2 37 47 07 02 2016 24 07 2016 Copyright © 2015, Yazd University. 2015 http://as.yazd.ac.ir/article_785.html

Assume that \$(N,L)\$, is a pair of finite dimensional nilpotent Lie algebras, in which \$L\$ is non-abelian and \$N\$ is an ideal in \$L\$ and also \$mathcal{M}(N,L)\$ is the Schur multiplier of the pair \$(N,L)\$. Motivated by characterization of the pairs \$(N,L)\$ of finite dimensional nilpotent Lie algebras by their Schur multipliers (Arabyani, et al. 2014) we prove some properties of a pair of nilpotent Lie algebras and generalize results for a pair of non-abelian nilpotent Lie algebras.

Nilpotent Lie algebra Pair of Lie algebras Schur multiplier