The variety generated by an ai-semiring of order three. (English) Zbl 1489.16050

Semirings are algebras \((S,+,\cdot)\) such that the additive reduct is a commutative semigroup, the multiplicative reduct is a semigroup, and where multiplication distributes over addition. A semiring is an ai-semiring if it is addditively idempotent, i.e,. its additive reduct is a semilattice. The authors continue their investigation of varieties of semirings started in many earlier publications. In this paper they are interested in the finite basis problem for the varieties of ai-semirings generated by a unique three element semiring. After an introduction recalling some known results concerning the finite basis problem, they note that there are precisely six two element ai-semirings, and the variety generated by any one of them is finitely based. Up to isomorphism there are \(61\) three element semirings. It is known from previous publications that each of certain \(45\) of them generate a finitely based variety. The authors focus their attention on the remaining varieties. The main results of this paper shows that with a possible exception of one variety, all of them are finitely based. The equational base for each of these varieties is provided. Then it is conjectured that the one remaining variety is not finitely based.


16Y60 Semirings
08B05 Equational logic, Mal’tsev conditions
Full Text: DOI MNR


[1] Dolinka I., “A nonfintely based finite semiring”, Int. J. Algebra Comput., 17:8 (2007), 1537-1551 · Zbl 1137.08003
[2] Dolinka I., “A class of inherently nonfinitely based semirings”, Algebra Universalis, 60:1 (2009), 19-35 · Zbl 1172.08003
[3] Dolinka I., “The finite basis problem for endomorphism semirings of finite semilattices with zero”, Algebra Universalis, 61:3-4 (2009), 441-448 · Zbl 1197.16047
[4] Dolinka I., “A remark on nonfinitely based semirings”, Semigroup Forum, 78:2 (2009), 368-373 · Zbl 1176.16038
[5] Ghosh S., Pastijn F., Zhao X.Ż., “Varieties generated by ordered bands I”, Order, 22:2 (2005), 109-128 · Zbl 1097.16018
[6] Kruse R. L., “Identities satisfied by a finite ring”, J. Algebra, 26:2 (1973), 298-318 · Zbl 0276.16014
[7] Kuřil M., Polák L., “On varieties of semilattice-ordered semigroups”, Semigroup Forum, 71:1 (2005), 27-48 · Zbl 1090.20028
[8] L’vov I. V., “Varieties of associative rings. I”, Algebra and Logic, 12:3 (1973), 150-167 · Zbl 0288.16008
[9] Lyndon R. C., “Identities in two-valued calculi.”, Trans. Amer. Math. Soc., 71:3 (1951), 457-457 · Zbl 0044.00201
[10] Lyndon R. C., “Identities in finite algebras”, Proc. Amer. Math. Soc., 5 (1954), 8-9 · Zbl 0055.02705
[11] McKenzie R., “Equational bases for lattice theories”, Math. Scand., 27 (1970), 24-38 · Zbl 0307.08001
[12] {McKenzie R.} Tarski’s finite basis problem is undecidable, Int. J. Algebra Comput., 6:1 (1996), 49-104 · Zbl 0844.08011
[13] McKenzie R. C., Romanowska A., “Varieties of \(\cdot \)-distributive bisemilattices”, Contrib. Gen. Algebra, 1 (1979), 213-218 · Zbl 0419.06003
[14] McNulty G. F., Willard R., The Chautauqua Problem, Tarski’s Finite Basis Problem, and Residual Bounds for 3-element Algebras (to appear)
[15] Oates S., Powell M. B., “Identical relations in finite groups”, J. Algebra, 1:1 (1964), 11-39 · Zbl 0121.27202
[16] Pastijn F., “Varieties generated by ordered bands II”, Order, 22:2 (2005), 129-143 · Zbl 1097.16019
[17] Pastijn F., Zhao X.Ż., “Varieties of idempotent semirings with commutative addition”, Algebra Universalis, 54:3 (2005), 301-321 · Zbl 1084.16039
[18] Perkins P., “Bases for equational theories of semigroups”, J. Algebra, 11:2 (1969), 298-314 · Zbl 0186.03401
[19] Ren M. M., Zhao X.Ż., “The varieties of semilattice-ordered semigroups satisfying \(x^3\approx x\) and \(xy\approx yx\)”, Period. Math. Hungar., 72:2 (2016), 158-170 · Zbl 1399.20062
[20] Ren M. M., Zhao X.Ż., Shao Y., “The lattice of ai-semiring varieties satisfying \(x^n\approx x\) and \(xy\approx yx\)”, Semigroup Forum, 100:2 (2020), 542-567 · Zbl 1457.16045
[21] Ren M. M., Zhao X.Ż., Wang A. F., “On the varieties of ai-semirings satisfying \(x^3\approx x\)”, Algebra Universalis, 77:4 (2017), 395-408 · Zbl 1371.08005
[22] Ren M. M., Zhao X.Ż., Volkov M. V., The Burnside Ai-Semiring Variety Defined by \(x^n\approx x\) (to appear)
[23] {Shao Y., Ren M. M.} On the varieties generated by ai-semirings of order two, Semigroup Forum, 91:1 (2015), 171-184 · Zbl 1347.16052
[24] Tarski A., “Equational logic and equational theories of algebras”, Stud. Logic Found. Math., 50 (1968), 275-288 · Zbl 0209.01402
[25] Vechtomov E. M., Petrov A. A., “Multiplicatively idempotent semirings”, J. Math. Sci., 206:6 (2015), 634-653 · Zbl 1333.16043
[26] Volkov M. V., “The finite basis problem for finite semigroups”, Sci. Math. Jpn., 53:1 (2001), 171-199 · Zbl 0990.20039
[27] Zhao X.Ż., Guo Y. Q., Shum K. P., “\( \mathcal{D} \)-subvarieties of the variety of idempotent semirings”, Algebra Colloquium, 9:1 (2002), 15-28 · Zbl 1004.16050
[28] Zhao X.Ż., Shum K. P., Guo Y. Q., “\( \mathcal{L} \)-subvarieties of the variety of idempotent semirings”, Algebra Universalis, 46:1-2 (2001), 75-96 · Zbl 1063.08009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.