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.  Google Scholar

[2]

K. AuingerI. DolinkaT. 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.  Google Scholar

[3]

K. AuingerI. Dolinka and M. V. Volkov, Matrix identities involving multiplication and transposition, J. Eur. Math. Soc., 14 (2012), 937-969.  doi: 10.4171/JEMS/323.  Google Scholar

[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.  Google Scholar

[5]

G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc., 31 (1935), 433-454.  doi: 10.1017/S0305004100013463.  Google Scholar

[6]

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer Verlag, New York, 1981.  Google Scholar

[7]

I. Dolinka, Remarks on varieties of involution bands, Comm. Algebra, 28 (2000), 2837-2852.  doi: 10.1080/00927870008826995.  Google Scholar

[8]

S. Fajtlowicz, Equationally complete semigroups with involution, Algebra Universalis, 1 (1971), 355-358.  doi: 10.1007/BF02944993.  Google Scholar

[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.  Google Scholar

[10]

J. Ježek, Nonfinitely based three-element idempotent groupoids, Algebra Universalis, 20 (1985), 292-301.  doi: 10.1007/BF01195139.  Google Scholar

[11]

E. I. Kleĭman, On basis of identities of Brandt semigroups, Semigroup Forum, 13 (1977), 209-218.  doi: 10.1007/BF02194938.  Google Scholar

[12]

E. I. Kleĭman, Bases of identities of varieties of inverse semigroups, (Russian) Sibirsk. Mat. Zh., 20 (1979), 760-777.  doi: 10.1007/BF00970367.  Google Scholar

[13]

R. L. Kruse, Identities satisfied by a finite ring, Algebra J., 26 (1973), 298-318.  doi: 10.1016/0021-8693(73)90025-2.  Google Scholar

[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.  Google Scholar

[15]

E. W. H. Lee, A class of finite semigroups without irredundant bases of identities, Yokohama Math. J., 61 (2015), 1-28.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

E. W. H. Lee, A sufficient condition for the absence of irredundant bases, to appear in Houston J. Math. Google Scholar

[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.  Google Scholar

[20]

E. W. H. LeeJ. R. Li and W. T. Zhang, Minimal non-finitely based semigroups, Semigroup Forum, 85 (2012), 577-580.  doi: 10.10107/s00233-012-9434-y.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. McKenzie, Equational bases for lattice theories, Math. Scand., 27 (1970), 24-38.  doi: 10.7146/math.scand.a-10984.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

P. Perkins, Bases for equational theories of semigroups, Algebra J., 11 (1969), 298-314.  doi: 10.1016/0021-8693(69)90058-1.  Google Scholar

[28]

P. Perkins, Finite axiomatizability for equational theories of computable groupoids, J. Symbolic Logic, 54 (1989), 1018-1022.  doi: 10.2307/2274762.  Google Scholar

[29]

M. V. Sapir, Inherently non-finitely based finite semigroups, (Russian) Mat. Sb. (N.S.), 133 (1987), 154-166.  doi: 10.1070/SM1988v061n01ABEH003199.  Google Scholar

[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.  Google Scholar

[31]

M. V. Sapir, Identities of finite inverse semigroups, Internat. J. Algebra Comput., 3 (1993), 115-124.  doi: 10.1142/S0218196793000093.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

M. V. Volkov, Bases of identities of Brandt semigroups, (Russian) Ural. Gos. Univ. Mat. Zap., 14 (1985), 38-42.   Google Scholar

[35]

M. V. Volkov, The finite basis question for varieties of semigroups, (Russian) Mat. Zametki, 45 (1989), 12-23.  doi: 10.1007/BF01158553.  Google Scholar

[36]

M. V. Volkov, The finite basis problem for finite semigroups, Sci. Math. Jpn., 53 (2001), 171-199.   Google Scholar

show all references

References:
[1]

J. Almeida, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994.  Google Scholar

[2]

K. AuingerI. DolinkaT. 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.  Google Scholar

[3]

K. AuingerI. Dolinka and M. V. Volkov, Matrix identities involving multiplication and transposition, J. Eur. Math. Soc., 14 (2012), 937-969.  doi: 10.4171/JEMS/323.  Google Scholar

[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.  Google Scholar

[5]

G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc., 31 (1935), 433-454.  doi: 10.1017/S0305004100013463.  Google Scholar

[6]

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer Verlag, New York, 1981.  Google Scholar

[7]

I. Dolinka, Remarks on varieties of involution bands, Comm. Algebra, 28 (2000), 2837-2852.  doi: 10.1080/00927870008826995.  Google Scholar

[8]

S. Fajtlowicz, Equationally complete semigroups with involution, Algebra Universalis, 1 (1971), 355-358.  doi: 10.1007/BF02944993.  Google Scholar

[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.  Google Scholar

[10]

J. Ježek, Nonfinitely based three-element idempotent groupoids, Algebra Universalis, 20 (1985), 292-301.  doi: 10.1007/BF01195139.  Google Scholar

[11]

E. I. Kleĭman, On basis of identities of Brandt semigroups, Semigroup Forum, 13 (1977), 209-218.  doi: 10.1007/BF02194938.  Google Scholar

[12]

E. I. Kleĭman, Bases of identities of varieties of inverse semigroups, (Russian) Sibirsk. Mat. Zh., 20 (1979), 760-777.  doi: 10.1007/BF00970367.  Google Scholar

[13]

R. L. Kruse, Identities satisfied by a finite ring, Algebra J., 26 (1973), 298-318.  doi: 10.1016/0021-8693(73)90025-2.  Google Scholar

[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.  Google Scholar

[15]

E. W. H. Lee, A class of finite semigroups without irredundant bases of identities, Yokohama Math. J., 61 (2015), 1-28.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

E. W. H. Lee, A sufficient condition for the absence of irredundant bases, to appear in Houston J. Math. Google Scholar

[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.  Google Scholar

[20]

E. W. H. LeeJ. R. Li and W. T. Zhang, Minimal non-finitely based semigroups, Semigroup Forum, 85 (2012), 577-580.  doi: 10.10107/s00233-012-9434-y.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. McKenzie, Equational bases for lattice theories, Math. Scand., 27 (1970), 24-38.  doi: 10.7146/math.scand.a-10984.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

P. Perkins, Bases for equational theories of semigroups, Algebra J., 11 (1969), 298-314.  doi: 10.1016/0021-8693(69)90058-1.  Google Scholar

[28]

P. Perkins, Finite axiomatizability for equational theories of computable groupoids, J. Symbolic Logic, 54 (1989), 1018-1022.  doi: 10.2307/2274762.  Google Scholar

[29]

M. V. Sapir, Inherently non-finitely based finite semigroups, (Russian) Mat. Sb. (N.S.), 133 (1987), 154-166.  doi: 10.1070/SM1988v061n01ABEH003199.  Google Scholar

[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.  Google Scholar

[31]

M. V. Sapir, Identities of finite inverse semigroups, Internat. J. Algebra Comput., 3 (1993), 115-124.  doi: 10.1142/S0218196793000093.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

M. V. Volkov, Bases of identities of Brandt semigroups, (Russian) Ural. Gos. Univ. Mat. Zap., 14 (1985), 38-42.   Google Scholar

[35]

M. V. Volkov, The finite basis question for varieties of semigroups, (Russian) Mat. Zametki, 45 (1989), 12-23.  doi: 10.1007/BF01158553.  Google Scholar

[36]

M. V. Volkov, The finite basis problem for finite semigroups, Sci. Math. Jpn., 53 (2001), 171-199.   Google Scholar

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