THE VARIETY GENERATED BY AN AI-SEMIRING OF ORDER THREE

Xianzhong Zhao     (Northwest University, Xi’an, 710069, P.R., China)
Miaomiao Ren     (Northwest University, Xi’an, 710069, P.R., China)
Siniša Crvenković     (University of Novi Sad, Novi Sad, 21102, Serbia)
Yong Shao     (Northwest University, Xi’an, 710069, P.R., China)
Petar Dapić     (University of Novi Sad, Novi Sad, 21102, Serbia)

Abstract


Up to isomorphism, there are 61 ai-semirings of order three. The finite basis problem for these semirings is investigated. This problem for 45 semirings of them is answered by some results in the literature. The remaining semirings are studied using equational logic. It is shown that with the possible exception of the semiring \(S_7\), all ai-semirings of order three are finitely based.


Keywords


Ai-semiring, Identity, Finitely based variety.

Full Text:

PDF

References


Dolinka I. A nonfintely based finite semiring. Int. J. Algebra Comput., 2007. Vol. 17, No. 8. P. 1537–1551. DOI: 10.1142/S0218196707004177

Dolinka I. A class of inherently nonfinitely based semirings. Algebra Universalis, 2009. Vol. 60, No. 1. P. 19–35. DOI: 10.1007/s00012-008-2084-y

Dolinka I. The finite basis problem for endomorphism semirings of finite semilattices with zero. Algebra Universalis, 2009. Vol. 61, No. 3–4. P. 441–448. DOI: 10.1007/s00012-009-0024-0

Dolinka I. A remark on nonfinitely based semirings. Semigroup Forum, 2009. Vol. 78, No. 2. P. 368–373. DOI: 10.1007/s00233-008-9096-y

Ghosh S., Pastijn F., Zhao X.Z. Varieties generated by ordered bands I. Order, 2005. Vol. 22, No. 2. P. 109–128. DOI: 10.1007/s11083-005-9011-z

Kruse R.L. Identities satisfied by a finite ring. J. Algebra, 1973. Vol. 26, No. 2. P. 298–318. DOI: 10.1016/0021-8693(73)90025-2

Kuřil M., Polák L. On varieties of semilattice-ordered semigroups. Semigroup Forum, 2005. Vol. 71, No. 1. P. 27–48. DOI: 10.1007/s00233-004-0176-3

L’vov I.V. Varieties of associative rings. I. Algebra and Logic, 1973. Vol. 12, No. 3. P. 150–167. DOI: 10.1007/BF02218695

Lyndon R.C. Identities in two-valued calculi. Trans. Amer. Math. Soc., 1951. Vol. 71, No. 3. P. 457–457. DOI: 10.1090/S0002-9947-1951-0044470-3

Lyndon R.C. Identities in finite algebras. Proc. Amer. Math. Soc., 1954. Vol. 5. P. 8–9. DOI: 10.1090/S0002-9939-1954-0060482-6

McKenzie R. Equational bases for lattice theories. Math. Scand., 1970. Vol. 27. P. 24–38. DOI: 10.7146/math.scand.a-10984

McKenzie R. Tarski’s finite basis problem is undecidable. Int. J. Algebra Comput., 1996. Vol. 6, No. 1. P. 49–104. DOI: 10.1142/S0218196796000040

McKenzie R.C., Romanowska A. Varieties of ·-distributive bisemilattices. Contrib. Gen. Algebra, 1979. Vol. 1. P. 213–218.

McNulty G.F., Willard R. The Chautauqua Problem, Tarski's Finite Basis Problem, and Residual Bounds for 3-element Algebras.

Oates S., Powell M.B. Identical relations in finite groups. J. Algebra, 1964. Vol. 1, No. 1. P. 11–39. DOI: 10.1016/0021-8693(64)90004-3

Pastijn F. Varieties generated by ordered bands II. Order, 2005. Vol. 22, No. 2. P. 129–143. DOI: 10.1007/s11083-005-9013-x

Pastijn F., Zhao X.Z. Varieties of idempotent semirings with commutative addition. Algebra Universalis, 2005. Vol. 54, No. 3. P. 301–321. DOI: 110.1007/s00012-005-1947-8

Perkins P. Bases for equational theories of semigroups. J. Algebra, 1969. Vol. 11, No. 2. P. 298–314. DOI: 10.1016/0021-8693(69)90058-1

Ren M.M., Zhao X.Z. The varieties of semilattice-ordered semigroups satisfying \(x^3\approx x\) and \(xy\approx yx\). Period. Math. Hungar., 2016. Vol. 72, No. 2. P. 158–170. DOI: 10.1007/s10998-016-0116-5

Ren M.M., Zhao X.Z., Shao Y. The lattice of ai-semiring varieties satisfying \(x^{n}\approx x\) and \(xy\approx yx\). Semigroup Forum, 2020. Vol. 100, No. 2. P. 542–567. DOI: 10.1007/s00233-020-10092-8

Ren M.M., Zhao X.Z., Wang A.F. On the varieties of ai-semirings satisfying \(x^{3}\approx x\). Algebra Universalis, 2017. Vol. 77, No. 4. P. 395–408. DOI: 10.1007/s00012-017-0438-z

Ren M.M., Zhao X.Z., Volkov M.V. The Burnside Ai-Semiring Variety Defined by \(x^{n}\approx x\). Manuscript.

Shao Y., Ren M.M. On the varieties generated by ai-semirings of order two. Semigroup Forum, 2015. Vol. 91, No. 1. P. 171–184. DOI: 10.1007/s00233-014-9667-z

Tarski A. Equational logic and equational theories of algebras. Stud. Logic Found. Math., 1968. Vol. 50. P. 275-288. DOI: 10.1016/S0049-237X(08)70531-7

Vechtomov E.M., Petrov A.A. Multiplicatively idempotent semirings. J. Math. Sci., 2015. Vol. 206, No. 6. P. 634–653. DOI: 10.1007/s10958-015-2340-6

Volkov M.V. The finite basis problem for finite semigroups. Sci. Math. Jpn., 2001. Vol. 53, No. 1. 171–199.

Zhao X.Z., Guo Y.Q., Shum K.P. \( \mathcal{D}\)-subvarieties of the variety of idempotent semirings. Algebra Colloquium, 2002. Vol. 9, No. 1. P. 15–28.

Zhao X.Z., Shum K.P., Guo Y.Q. \(\mathcal{L}\)-subvarieties of the variety of idempotent semirings. Algebra Universalis, 2001. Vol. 46, No. 1-2. P. 75–96. DOI: 10.1007/PL00000348




DOI: http://dx.doi.org/10.15826/umj.2020.2.012

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.