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S111 from the proof of Proposition 4, and therefore, the variety var UT± 3 (R) satisfies the condition (i ) of the stronger form of Theorem 1. Of course, the variety fulfills also the condition (ii) since (ii) is inherited by supervarieties. In the same fashion, the proof of Theorem 2 applies, say, to the semigroup of all upper triangular complex 3 × 3-matrices whose main diagonal entries come from the set {0, 1, ξ, . . , ξ n−1 } where ξ is a primitive nth root of unity. The question of whether or not a result similar to Theorem 2 holds true for analogs of the semigroup UT3 (R) in other dimensions is more involved.

30, 295–314 (1987)] 18. : The finite basis problem for finite semigroups, Sci. Math. Jpn. 53,171–199 (2001). ] 19. : Semigroups that are nilpotent in the sense of Mal’cev. Izv. Vyssh. Uchebn. Zaved. 6, 23–29 (1980). S. Nambooripad Abstract The semigroup of all linear maps on a vector space is regular, but the semigroup of continuous linear maps on a Hilbert space is not, in general, regular; nor is the product of two regular elements regular. In this chapter, we show that in those types of von Neumann algebras of operators in which the lattice of projections is modular, the set of regular elements do form a (necessarily regular) semigroup.

Proposition 3 For an operator v on H , the following are equivalent (i) (ii) (iii) (iv) v is a partial isometry vv ∗ is a projection v ∗ v is a projection vv ∗ v = v Now from (iv) above, we have v ∗ = (vv ∗ v)∗ = v ∗ v ∗∗ v ∗ = v ∗ vv ∗ and this together with (iv) shows that v ∗ is an inverse of v. In view of (ii) and (iii), it follows that the adjoint v ∗ is in fact the Moore–Penrose inverse of v. Thus we can add the following characterization to the above list: Proposition 4 An operator v on H is a partial isometry if and only if v is a regular element of B(H ) for which v † = v ∗ .

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