Document Type : Research Paper

**Authors**

Department of Mathematics, Tehran Science and Research Branch, Islamic Azad University, P.O. Box 14515/1775, Tehran, Iran.

**Abstract**

Decomposability of an algebraic structure into the union of its sub-structures goes back to G. Scorza's Theorem of 1926 for groups. An analogue of this theorem for rings has been recently studied by A. Lucchini in 2012. On the study of this problem for non-group semigroups, the first attempt is due to Clifford's work of 1961 for the regular semigroups. Since then, N.P. Mukherjee in 1972 studied the decomposition of quasicommutative semigroups where, he proved that: a regular quasicommutative semigroup is decomposable into the union of groups. The converse of this result is a natural question. Obviously, if a semigroup $S$ is decomposable into a union of groups then $S$ is regular so, the aim of this paper is to give examples of non-quasicommutative semigroups which are decomposable into the disjoint unions of

groups. Our examples are the semigroups presented by the following presentations: $$\pi_1 =\langle a,b\mid a^{n+1}=a, b^3=b, ba=a^{n-1}b\rangle,~(n\geq 3),$$

$$\pi_2 =\langle a,b\mid a^{1+p^\alpha}=a, b^{1+p^\beta}=b, ab=ba^{1+p^{\alpha-\gamma}}\rangle$$where, $p$ is an odd prime, $\alpha, \beta$ and $\gamma$ are integers such that $\alpha \geq 2\gamma$, $\beta \geq \gamma \geq 1$ and $\alpha +\beta > 3$.

groups. Our examples are the semigroups presented by the following presentations: $$\pi_1 =\langle a,b\mid a^{n+1}=a, b^3=b, ba=a^{n-1}b\rangle,~(n\geq 3),$$

$$\pi_2 =\langle a,b\mid a^{1+p^\alpha}=a, b^{1+p^\beta}=b, ab=ba^{1+p^{\alpha-\gamma}}\rangle$$where, $p$ is an odd prime, $\alpha, \beta$ and $\gamma$ are integers such that $\alpha \geq 2\gamma$, $\beta \geq \gamma \geq 1$ and $\alpha +\beta > 3$.

**Keywords**

**Main Subjects**

A. Arjomandfar, C.M. Campbell and H. Doostie, Semigroups related to certain group presentations, *Semigroup Forum ***85 **(2012), no. 3, 533--539.

C. M. Campbell, E. F. Robertson, N. Ruskuc and R. M. Thomas, Semigroup and group presentations, *Bull. London Math. Soc. ***27 **(1995), no. 1, 46--50.

A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, I, Amer. Math. Soc. Surveys 7, Providence, 1961.

J. Howie, Fundamentals of Semigroup Theory, London Mathematical Society, New Series, Oxford University Press, New York, 1995.

A. Lucchini and A. Maroti, Rings as the union of proper subrings, *Algebr. Represent. Theory ***15 **(2012), no. 6, 1035--1047.

N. P. Mukherjee, Quasicommutative semigroups I, *Czechoslovak Math. J. ***22(97) **(1972) 449--453.

E. F. Robertson and Y. Ünlü, On semigroup presentations, *Proc. Edinburgh Math. Soc. (2) ***36 **(1993), no. 1, 55--68.

G. Scorza, I gruppi che possono pensarsi come somma di tre loro sottogruppi, *Boll. Un. Mat. Ital. ***5 **(1926) 216--218.

April 2016

Pages 483-487

**Receive Date:**05 June 2014**Revise Date:**22 February 2015**Accept Date:**23 February 2015