# American Institute of Mathematical Sciences

2017, 24: 10-20. doi: 10.3934/era.2017.24.002

## Equational theories of unstable involution semigroups

 Department of Mathematics, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, Florida 33314, USA

The author is indebted to the referee for insightful comments and a thorough review. Results of the present article were announced in Workshop on Groups and Semigroups: on the occasion of the 60th birthday of Mikhail Volkov held at the University of Porto on June 9,2015

Received  December 08, 2016 Published  March 2017

It is long known that with respect to the property of having a finitely axiomatizable equational theory, there is no relationship between a general involution semigroup and its semigroup reduct. The present article establishes such a relationship within the class of involution semigroups that are unstable in the sense that the varieties they generate contain semilattices with nontrivial involution. Specifically, it is shown that the equational theory of an unstable involution semigroup is not finitely axiomatizable whenever the equational theory of its semigroup reduct satisfies the same property. Consequently, many results on equational properties of semigroups can be converted into results applicable to involution semigroups.

Citation: Edmond W. H. Lee. Equational theories of unstable involution semigroups. Electronic Research Announcements, 2017, 24: 10-20. doi: 10.3934/era.2017.24.002
##### References:
 [1] J. Almeida, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994. [2] K. Auinger, I. Dolinka, T. V. Pervukhina and M. V. Volkov, Unary enhancements of inherently non-finitely based semigroups, Semigroup Forum, 89 (2014), 41-51.  doi: 10.1007/s00233-013-9509-4. [3] K. Auinger, I. Dolinka and M. V. Volkov, Matrix identities involving multiplication and transposition, J. Eur. Math. Soc., 14 (2012), 937-969.  doi: 10.4171/JEMS/323. [4] Yu. A. Bahturin and A. Yu. Ol'shanskiĭ, Identical relations in finite Lie rings, (Russian) Mat. Sb. (N.S.), 96 (1975), 543-559.  doi: 10.1070/SM1975v025n04ABEH002459. [5] G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc., 31 (1935), 433-454.  doi: 10.1017/S0305004100013463. [6] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer Verlag, New York, 1981. [7] I. Dolinka, Remarks on varieties of involution bands, Comm. Algebra, 28 (2000), 2837-2852.  doi: 10.1080/00927870008826995. [8] S. Fajtlowicz, Equationally complete semigroups with involution, Algebra Universalis, 1 (1971), 355-358.  doi: 10.1007/BF02944993. [9] M. Jackson and M. V. Volkov, The algebra of adjacency patterns: Rees matrix semigroups with reversion, in Fields of Logic and Computation, Lecture Notes in Comput. Sci., 6300, Springer, Berlin, 2010,414-443. doi: 10.1007/978-3-642-15025-8_20. [10] J. Ježek, Nonfinitely based three-element idempotent groupoids, Algebra Universalis, 20 (1985), 292-301.  doi: 10.1007/BF01195139. [11] E. I. Kleĭman, On basis of identities of Brandt semigroups, Semigroup Forum, 13 (1977), 209-218.  doi: 10.1007/BF02194938. [12] E. I. Kleĭman, Bases of identities of varieties of inverse semigroups, (Russian) Sibirsk. Mat. Zh., 20 (1979), 760-777.  doi: 10.1007/BF00970367. [13] R. L. Kruse, Identities satisfied by a finite ring, Algebra J., 26 (1973), 298-318.  doi: 10.1016/0021-8693(73)90025-2. [14] E. W. H. Lee, Finite basis problem for semigroups of order five or less: Generalization and revisitation, Studia Logica, 101 (2013), 95-115.  doi: 10.1007/s11225-012-9369-z. [15] E. W. H. Lee, A class of finite semigroups without irredundant bases of identities, Yokohama Math. J., 61 (2015), 1-28. [16] E. W. H. Lee, Finite involution semigroups with infinite irredundant bases of identities, Forum Math., 28 (2016), 587-607.  doi: 10.1515/forum-2014-0098. [17] E. W. H. Lee, Finitely based finite involution semigroups with non-finitely based reducts, Quaest. Math., 39 (2016), 217-243.  doi: 10.2989/16073606.2015.1068239. [18] E. W. H. Lee, A sufficient condition for the absence of irredundant bases, to appear in Houston J. Math. [19] E. W. H. Lee and J. R. Li, Minimal non-finitely based monoids, Dissertationes Math. (Rozprawy Mat.), 475 (2011), 65.  doi: 10.4064/dm475-0-1. [20] E. W. H. Lee, J. R. Li and W. T. Zhang, Minimal non-finitely based semigroups, Semigroup Forum, 85 (2012), 577-580.  doi: 10.10107/s00233-012-9434-y. [21] E. W. H. Lee and W. T. Zhang, Finite basis problem for semigroups of order six, LMS J. Comput. Math., 18 (2015), 1-129.  doi: 10.1112/S1461157014000412. [22] I. V. L'vov, Varieties of associative rings. Ⅰ., (Russian) Algebra i Logika, 12 (1973), 269-297; translation in Algebra and Logic, 12 (1973), 667-688. doi: 10.1007/BF02218695. [23] R. McKenzie, Equational bases for lattice theories, Math. Scand., 27 (1970), 24-38.  doi: 10.7146/math.scand.a-10984. [24] I. I. Mel'nik, Varieties and lattices of varieties of semigroups, (Russian) in Studies in Algebra, No. 2, Izdat. Saratov. Univ., Saratov, 1970, 47-57. [25] V. L. Murskiĭ, The existence in the three-valued logic of a closed class with a finite basis, having no finite complete system of identities, (Russian) Dokl. Akad. Nauk SSSR, 163 (1965), 815-818. [26] S. Oates and M. B. Powell, Identical relations in finite groups, Algebra J., 1 (1964), 11-39.  doi: 10.1016/0021-8693(64)90004-3. [27] P. Perkins, Bases for equational theories of semigroups, Algebra J., 11 (1969), 298-314.  doi: 10.1016/0021-8693(69)90058-1. [28] P. Perkins, Finite axiomatizability for equational theories of computable groupoids, J. Symbolic Logic, 54 (1989), 1018-1022.  doi: 10.2307/2274762. [29] M. V. Sapir, Inherently non-finitely based finite semigroups, (Russian) Mat. Sb. (N.S.), 133 (1987), 154-166.  doi: 10.1070/SM1988v061n01ABEH003199. [30] M. V. Sapir, Problems of Burnside type and the finite basis property in varieties of semigroups, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 319-340.  doi: 10.1070/IM1988v030n02ABEH001012. [31] M. V. Sapir, Identities of finite inverse semigroups, Internat. J. Algebra Comput., 3 (1993), 115-124.  doi: 10.1142/S0218196793000093. [32] A. Tarski, Equational logic and equational theories of algebras, in Contributions to Mathematical Logic (Hannover, 1966) (eds. H. A. Schmidt, et al.), North-Holland, Amsterdam, 1968,275-288. doi: 10.1016/S0049-237X(08)70531-7. [33] A. N. Trahtman, The finite basis question for semigroups of order less than six, Semigroup Forum, 27 (1983), 387-389.  doi: 10.1007/BF02572749. [34] M. V. Volkov, Bases of identities of Brandt semigroups, (Russian) Ural. Gos. Univ. Mat. Zap., 14 (1985), 38-42. [35] M. V. Volkov, The finite basis question for varieties of semigroups, (Russian) Mat. Zametki, 45 (1989), 12-23.  doi: 10.1007/BF01158553. [36] M. V. Volkov, The finite basis problem for finite semigroups, Sci. Math. Jpn., 53 (2001), 171-199.

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##### References:
 [1] J. Almeida, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994. [2] K. Auinger, I. Dolinka, T. V. Pervukhina and M. V. Volkov, Unary enhancements of inherently non-finitely based semigroups, Semigroup Forum, 89 (2014), 41-51.  doi: 10.1007/s00233-013-9509-4. [3] K. Auinger, I. Dolinka and M. V. Volkov, Matrix identities involving multiplication and transposition, J. Eur. Math. Soc., 14 (2012), 937-969.  doi: 10.4171/JEMS/323. [4] Yu. A. Bahturin and A. Yu. Ol'shanskiĭ, Identical relations in finite Lie rings, (Russian) Mat. Sb. (N.S.), 96 (1975), 543-559.  doi: 10.1070/SM1975v025n04ABEH002459. [5] G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc., 31 (1935), 433-454.  doi: 10.1017/S0305004100013463. [6] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer Verlag, New York, 1981. [7] I. Dolinka, Remarks on varieties of involution bands, Comm. Algebra, 28 (2000), 2837-2852.  doi: 10.1080/00927870008826995. [8] S. Fajtlowicz, Equationally complete semigroups with involution, Algebra Universalis, 1 (1971), 355-358.  doi: 10.1007/BF02944993. [9] M. Jackson and M. V. Volkov, The algebra of adjacency patterns: Rees matrix semigroups with reversion, in Fields of Logic and Computation, Lecture Notes in Comput. Sci., 6300, Springer, Berlin, 2010,414-443. doi: 10.1007/978-3-642-15025-8_20. [10] J. Ježek, Nonfinitely based three-element idempotent groupoids, Algebra Universalis, 20 (1985), 292-301.  doi: 10.1007/BF01195139. [11] E. I. Kleĭman, On basis of identities of Brandt semigroups, Semigroup Forum, 13 (1977), 209-218.  doi: 10.1007/BF02194938. [12] E. I. Kleĭman, Bases of identities of varieties of inverse semigroups, (Russian) Sibirsk. Mat. Zh., 20 (1979), 760-777.  doi: 10.1007/BF00970367. [13] R. L. Kruse, Identities satisfied by a finite ring, Algebra J., 26 (1973), 298-318.  doi: 10.1016/0021-8693(73)90025-2. [14] E. W. H. Lee, Finite basis problem for semigroups of order five or less: Generalization and revisitation, Studia Logica, 101 (2013), 95-115.  doi: 10.1007/s11225-012-9369-z. [15] E. W. H. Lee, A class of finite semigroups without irredundant bases of identities, Yokohama Math. J., 61 (2015), 1-28. [16] E. W. H. Lee, Finite involution semigroups with infinite irredundant bases of identities, Forum Math., 28 (2016), 587-607.  doi: 10.1515/forum-2014-0098. [17] E. W. H. Lee, Finitely based finite involution semigroups with non-finitely based reducts, Quaest. Math., 39 (2016), 217-243.  doi: 10.2989/16073606.2015.1068239. [18] E. W. H. Lee, A sufficient condition for the absence of irredundant bases, to appear in Houston J. Math. [19] E. W. H. Lee and J. R. Li, Minimal non-finitely based monoids, Dissertationes Math. (Rozprawy Mat.), 475 (2011), 65.  doi: 10.4064/dm475-0-1. [20] E. W. H. Lee, J. R. Li and W. T. Zhang, Minimal non-finitely based semigroups, Semigroup Forum, 85 (2012), 577-580.  doi: 10.10107/s00233-012-9434-y. [21] E. W. H. Lee and W. T. Zhang, Finite basis problem for semigroups of order six, LMS J. Comput. Math., 18 (2015), 1-129.  doi: 10.1112/S1461157014000412. [22] I. V. L'vov, Varieties of associative rings. Ⅰ., (Russian) Algebra i Logika, 12 (1973), 269-297; translation in Algebra and Logic, 12 (1973), 667-688. doi: 10.1007/BF02218695. [23] R. McKenzie, Equational bases for lattice theories, Math. Scand., 27 (1970), 24-38.  doi: 10.7146/math.scand.a-10984. [24] I. I. Mel'nik, Varieties and lattices of varieties of semigroups, (Russian) in Studies in Algebra, No. 2, Izdat. Saratov. Univ., Saratov, 1970, 47-57. [25] V. L. Murskiĭ, The existence in the three-valued logic of a closed class with a finite basis, having no finite complete system of identities, (Russian) Dokl. Akad. Nauk SSSR, 163 (1965), 815-818. [26] S. Oates and M. B. Powell, Identical relations in finite groups, Algebra J., 1 (1964), 11-39.  doi: 10.1016/0021-8693(64)90004-3. [27] P. Perkins, Bases for equational theories of semigroups, Algebra J., 11 (1969), 298-314.  doi: 10.1016/0021-8693(69)90058-1. [28] P. Perkins, Finite axiomatizability for equational theories of computable groupoids, J. Symbolic Logic, 54 (1989), 1018-1022.  doi: 10.2307/2274762. [29] M. V. Sapir, Inherently non-finitely based finite semigroups, (Russian) Mat. Sb. (N.S.), 133 (1987), 154-166.  doi: 10.1070/SM1988v061n01ABEH003199. [30] M. V. Sapir, Problems of Burnside type and the finite basis property in varieties of semigroups, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), 319-340.  doi: 10.1070/IM1988v030n02ABEH001012. [31] M. V. Sapir, Identities of finite inverse semigroups, Internat. J. Algebra Comput., 3 (1993), 115-124.  doi: 10.1142/S0218196793000093. [32] A. Tarski, Equational logic and equational theories of algebras, in Contributions to Mathematical Logic (Hannover, 1966) (eds. H. A. Schmidt, et al.), North-Holland, Amsterdam, 1968,275-288. doi: 10.1016/S0049-237X(08)70531-7. [33] A. N. Trahtman, The finite basis question for semigroups of order less than six, Semigroup Forum, 27 (1983), 387-389.  doi: 10.1007/BF02572749. [34] M. V. Volkov, Bases of identities of Brandt semigroups, (Russian) Ural. Gos. Univ. Mat. Zap., 14 (1985), 38-42. [35] M. V. Volkov, The finite basis question for varieties of semigroups, (Russian) Mat. Zametki, 45 (1989), 12-23.  doi: 10.1007/BF01158553. [36] M. V. Volkov, The finite basis problem for finite semigroups, Sci. Math. Jpn., 53 (2001), 171-199.
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