Specht property for some varieties of Jordan algebras of almost polynomial growth

Let F be a field of characteristic zero. In [25] it was proved that UJ(2), the Jordan algebra of 2 x 2 upper triangular matrices, can be endowed up to isomorphism with either the trivial grading or three distinct non-trivial Z(2)-gradings or by a Z(2) x Z(2)-grading. In this paper we prove that the variety of Jordan algebras generated by UJ(2) endowed with any G-grading has the Specht property, i.e., every T-G-ideal containing the graded identities of UJ(2) is finitely based. Moreover, we prove an analogue result about the ordinary identities of A(1), a suitable infinitely generated metabelian Jordan algebra defined in [27]. (C) 2018 Elsevier Inc. All rights reserved.