SKEW NEAR-LATTICES IN RICKART RINGS J anis C rulis University - - PowerPoint PPT Presentation

SKEW NEAR-LATTICES IN RICKART RINGS

J¯ anis C ¯ ırulis

University of Latvia

email: jc@lanet.lv

• 88. Arbeitstagung Allgemeine Algebra (AAA88)

Warszawa, June 19 - 22, 2014

OVERVIEW

• 1. Basic definitions and facts

(skew near-lattices, examples, Rickart rings)

• 2. Rickart rings as skew near-lattices

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• 1. BASIC DEFINITIONS AND FACTS

1.1 Skew nearlattices

[J. C ¯ ırulis, Skew nearlattices: some structure and representation theorems,

• Contrib. General Algebra 19 (2010), 33–44]

The natural order ≤ in a band (A, △) is defined by x ≤ y iff y △ x = x = x △ y. A skew nearlattice is a band A which has the upper bound prop- erty under the natural ordering. Let ∨ and ∧ stand for the partial lattice operations in A under ≤.

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A particular class of skew nearlattices is that in which △ is com- mutative in every initial segment of A. This is the case if and only if the underlying band is normal, i.e., satisfies the identity: u △ x △ y △ v = u △ y △ x △ v, and then every initial segment [0, p] is a lattice with a ∧ b = a △ b.

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(Recall: x ≤ y iff y △ x = x = x △ y.)

The skew nearlattices arising in some applications are right-

(or left-) handed in the following sense:

x = x △ y implies that y △ x = x (i.e., if x ≤ y iff x △ y = x). This is the case if and only if the underlying band is right regular, i.e., satisfies the identity x △ y △ x = y △ x. A normal right-handed band is right normal (and conversely), i.e., satisfies the identity x △ y △ z = y △ x △ z.

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1.2. Three examples of right normal skew nearlattices

The set of partial functions X → V with △ defined by

φ △ ψ := ψ|(dom φ ∩ dom ψ).

The set of all subsets of X × Y (i.e., binary relations) with △

defined by R △ S := S ◦ (IR ∩ IS), where ◦ is composition, IR is the identity relation restricted to the range of R, and likewise IS.

The set of self-adjoint operators of a Hilbert space H with △

defined by A △ B := B max(P ∈ P: P ≤ PA ∧ PB, BP = PB), where P is the lattice of projection operators in H, and PC is the projection operator onto the closed range of C. [ A △ B := B(PA ∧ PB) for all bounded operators ]

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1.3. Rickart rings A (right) Rickart ring is an associative ring R such that the right annihilator of every element of R is a principal right ideal generated by an idempotent. A Rickart *-ring is an involution ring R which is Rickart and every each generating idempotent in the previous definition is symmetric (alias self-adjoint). We shall deal with Rickart rings which have certain involution- free properties of Rickart *-rings without introducing any invo- lution.

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Let R be a Rickart ring. This means that given x ∈ R, we can choose an idempotent x′ such that

✎ ✍ ☞ ✌

for all y ∈ R, xy = 0 iff x′y = y. We may assume that here

☛ ✡ ✟ ✠

x′′ = 1 − x′ , where 1 := 0′.

1 is the unit of R

This identity is equivalent to the condition

✎ ✍ ☞ ✌

for all y ∈ R, xy = 0 iff x′′y = 0. Idempotents of R in the range of the operation ′ are said to be closed. We denote the set of all such idempotents by P. Rickart rings, considered as algebras (R, +, ·, ′, 0), form a variety.

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Recall that idempotents of any ring form an orthomodular poset under the

• rdering

e ≤ f iff fe = e = ef.

We call a right Rickart ring strong if e ≤ f iff ef = e for all closed idempotents e and f. Proposition If R is strong, then P is an orthomodular lattice. In particular, every initial segment [0, g] of P is

a sublattice of P,

• rthomodular, with orthocomplementation ⊥

g given by

e⊥

g := g − e.

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• 2. A RICKART RING AS A SKEW NEARLATTICE

R – a strong Rickart ring. Let x △ y := y(x′′ ∧ y′′) (the skew meet of x an y). Theorem 1 (a) The algebra (R, △) is a right normal skew nearlattice with 0 the least element. (b) The partial join operation ∨ in R is given by if a, b ≤ x, then a ∨ b = x(a′′ ∨ b′′). (c) The lattice [0, 1] coincides with P. If the ring R happens to be regular, then the order ≤ agrees with the so called right star order defined by x ≤∗ y iff x = yx′′ and Rx ⊆ Ry.

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Theorem 2 (a) The mapping φ: x → x′′ is an order homomorfism of R

• nto P; moreover,

φ(x △ y) = φ(x) ∧ φ(y). (b) For every x ∈ R, the restriction of φ to [0, x] is an order isomorphism between this segment and [0, x′′]. (c) The kernel equivalence of φ is the Green’s equivalence R (and even D) of the band (R, △). Corollary (a) Every initial segment of R is an orthomodular lattice. (b) If x R y, then the segments [0, x] and [0, y] are order iso- morphic, the isomorphism being given by a → ya′′.

(a ∈ [0, x])

(c) Conversely, if these segments are order isomorphic, then x R y.

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