The class of all semigroups related to semihypergroups of order 2
-
Somnuek Worawiset
,
Jörg Koppitz
Abstract
This paper deals with semihypergroups of order two from the point of view of the model theory. We use basic knowledge to show that there are exactly 17 non-isomorphic semihypergroups of order two. Each of them corresponds in a canonical way to a semigroup of order three. We classify all of them by generalized identities a concept introduced by Lyapin. In particular, we classify all non-group semigroups of order three by one generalized identity.
(Communicated by Miroslav Ploščica)
Acknowledgement
The authors would like to express their gratitude to the referees for their helpful suggestions and comments. The first author is supported by the Thai Visiting Scholar 2017, Khon Kaen University, Thailand.
References
[1] Araújo, J. P.—Kinyon, M.: A natural chracterization of semilattices of rectangular bands and groups of exponent two, Semigroup Forum 91(1) (2015), 295–298.10.1007/s00233-014-9644-6Search in Google Scholar
[2] Bonansinga, P.—Corsini, P.: On semihypergroup and hypergroup homomorphisms, Unione Mat. Ital. Boll. B (6) 1(2) (1982), 717–727, (in Italian).Search in Google Scholar
[3] Changphas, T.—Davvaz, B.: Properties of Hyperideals in Ordered semigroups, Ital. J. Pure Appl. Math. 33 (2014), 425–432.Search in Google Scholar
[4] Corsini, P.: Sur les semi-hypergroupes, Atti Soc. Peloritana Sci. Fis. Mat. Natur. 26 (1980), 363–372, (in French).Search in Google Scholar
[5] Davvaz, B.: Semihypergroup Theory, Elsvier, Sci. Publication, 2016.10.1016/B978-0-12-809815-8.00002-4Search in Google Scholar
[6] Evseev, A. E.: Semigroups with some power identity inclusions, Algebraic systems with one operation and one relation. Interuniv. Collect. Sci. Works, Leningrad, 1985, pp. 21–32, (in Russian).Search in Google Scholar
[7] Freni, D.: Minimal order semihypergroups of type U on the right, II, J. Algebra 340 (2011), 77–89.10.1016/j.jalgebra.2011.05.015Search in Google Scholar
[8] Golmohamadian, M.—Zahedi, M. M.: HyperK-algebras induced by a deterministic finite automaton, Ital. J. Pure Appl. Math. 27 (2010), 119–140.Search in Google Scholar
[9] Hebisch, U.—Weinert, H. J.: Semirings – Algebraic Theory and Applications in Computer Science, World Scientific, Singapore, 1998.10.1142/3903Search in Google Scholar
[10] Hila, K.—Davvaz, B.—Naka, K.: On Quasi-hyperideals in Semihypergroups, Comm. Algebra 39(1) (2011), 4183–4194.10.1080/00927872.2010.521932Search in Google Scholar
[11] Kehayopolu, N.: On hypersemigroups, Pure Math. Appl. 25(2) (2015), 151–156.10.1515/puma-2015-0015Search in Google Scholar
[12] Kudryavtseva, G.—Mazorchuk, V.: On multisemigroups. Port. Math. (N.S.) 72(1) (2015), 47–80.10.4171/PM/1956Search in Google Scholar
[13] Leoreanu, V.: About the simplifiable cyclic semihypergroups, Ital. J. Pure Appl. Math. 7 (2000), 69–76.Search in Google Scholar
[14] Lyapin, E. S.: Identities valid globally in semigroups, Semigroup Forum 24(1) (1982), 263–269.10.1007/BF02572772Search in Google Scholar
[15] Marty, F.: Sur une generalization de la notion de groupe. 8iem Congres des Math. Scandinaves, Stockholm, 1934, pp. 45–49.Search in Google Scholar
[16] Monzo, R. A. R.: Semilattices of rectangular bands and groups of order two, aiXiv: 1301.0828 (2013).Search in Google Scholar
[17] Salvo, M. D.—Freni, D.—Faro, L. G.: Fully simple semihypergroups, J. Algebra 399 (2014), 358–377.10.1016/j.jalgebra.2013.09.046Search in Google Scholar
[18] Thron, R.—Koppitz, J.: Finite relational disjunctions, Algebra Colloq. 6(3) (1999), 261–268.Search in Google Scholar
[19] Zahedi, M. M.—Torkzaden, L.—Borzooei, R. A.: Hyper I-algebras and polygroups, Quasigroups Related Systems 11 (2004), 103–113.Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- RNDr. Miloslav Duchoň, DrSc. passed away ∗ aug. 21, 1932 – †dec. 31, 2018
- Prof. RNDr. Gejza Wimmer, DrSc. – A Septuagenerian?
- Two monads on the category of graphs
- Entropy of MV-algebraic dynamical systems: An example
- Going up and lying over in congruence-modular algebras
- The minimal arity of near unanimity polymorphisms
- On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one
- The generalized Fermat Conjecture
- Evaluation of sums involving products of Gaussian q-binomial coefficients with applications
- Integrals of logarithmic functions and alternating multiple zeta values
- An abelian subextension of the dyadic division field of a hyperelliptic Jacobian
- The class of all semigroups related to semihypergroups of order 2
- Criterion for the weak potency of certain HNN extensions
- Fekete Szegö theorem for a close-to-convex error function
- Bounds for distance between eigenvalues of boundary value problems with retarded argument
- Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow
- Weak module amenability of triangular banach algebras II
- Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class
- Example of C-rigid polytopes which are not B-rigid
- Some comments on rU-spaces
- On the equivalence of various definitions of mixed poisson processes
- A note on discrete C-embedded subspaces
