Structure of archimedean f-rings

A lattice-ordered ring, denoted ℓ-ring, is a ring R whose underlying set is lattice-ordered such that (R, ^ , +) is an l-group and such that if a ^ b and 0 ^ c in R, then ac ^ bc and ca ^ cb. An l-algebra is an algebra over the reals whose underlying vector space and ring are vector lattice and ℓ-ring, respectively. An ℓ-ring (ℓ-algebra) R is called an f -ring (f -algebra) if x Л y = 0 implies ax Л y = 0 and xa Л y = 0 for all x,y E E and a E R + (see [1, 2]). By erased vector lattices (erased f -algebras) we mean the ℓ -groups ( f -rings) that result from vector lattices ( f -algebras) by ignoring the multiplication by real numbers, see [4].

Say that the elements x, y E G of an l-group are disjoint and write x X y if | x | Л | y | = 0. For a nonempty subset M C G the polar M ^ is defined as M ^ := { x E G : ( V y E M ) x X y } . The inclusion ordered set of all polars in G is a complete Boolean algebra denoted by P(G) := (P(G), V , Л , * ), where L Л M = L П M , L V M = (L U M ) ^±± , L * = L ^ (L, M E P(G). If for every polar L E P(G) there is a polar decomposition G = L ® L ^ then G is called strongly projectable ; polar projections form a complete Boolean algebra which is isomorphic to P(G) and denoted by the same symbol, see [1, 2] for more details.

For a complete Boolean algebra B, denote by V ^(B) the corresponding Boolean valued model of set theory, [3, chap. 2]. There is a natural way of assigning to each statement φ about xi,..., x _n E V ^(B) the Boolean truth-value [ ф(x1,..., x _n )] E B. The sentence ф(x1,..., x _n ) is called true within V ^(B) if [ ф(x ₁ ,... , x _n )] = 1. For every complete Boolean algebra B, all the theorems of ZFC are true in V ^(B) . The ascending-and-descending machinery providing an

• ^#    The research is supported by the Ministry of Science and Higher Education of the Russian Federation, agreement № 075-02-2021-1844.

interaction between the Boolean-valued universe V ^(B) and the world of ordinary sets, rests on the functors of canonical embedding X H- X ^л and ascent X H- X f , both acting from V to V ^(B) , and the functor of descent X H- X ^ , acting from V ^(B) to V; see [3] for details. Boolean valued technology enables one to reduce some problems concerning ℓ-groups and f -rings to that of linearly ordered groups and rings, see [3, § 4.4]. The aim of this note is to outline a Boolean values approach to the structure theory of f -rings. Denote by R the fields of reals within V ^(B) . Then the descent R ^ is a universally complete vector lattice, [3, 5.2.2]. Below, Z will denote the integers and R the reals, both with the usual addition, multiplication, and order; denote by Z _a and R _a the additive grope of Z and the ring of R, respectively. Note that Z ^л is the ring of integers within V ^(B) and (Z _a ) ^л = (Z ^л ) _a , whilst the equality R ^л = R amounts to the σ-distributivity of B.

Lemma 1. Let K be a nonzero order complete f -ring, B = P(K) and K the Boolean valued representation of K in the model V ^(B) . Then V ^(B) = “ K is either the group of integers with zero multiplication Z _a ^∧ , or the ring of integers Z ^∧ , or the additive groups of reals with zero multiplication R _a , or the ring of reals R ”.

• < 1 Sketch Of the Proof. According to [3, Theorem 4.4.13] K is either a subgroup Ko of R a (with zero multiplication) or a subring K of R . Since K is order complete, so is K by [3, Theorem 4.4.10(2)]. It follows that the subgroup Ko is either Z a or is R a , whilst the subring K is either Z or is R . >

Lemma 2. Let K is an Archimedean f -ring and K be its Boolean valued representation of K ⁵ . There are a Boolean isomorphism i from B onto P(K ⁵ ) and a complete monomorphism j from K into K ‘ := K ^ such that b C [ 0 C j(g )] ^^ 0 C i(b)j(g) for all g E G и b E B .

• <    Lemma 1 is applicable to the Dedekind completion K ⁵ . Combining this with [3, Theorem 4.4.10] we arrive at the desired result. >

A ℓ -ring K is said to be lateral ly complete if each its disjoint subset has the least upper bound. Each ℓ -ring K has a uniquely determined lateral completion denoted by K ^λ , [4]. Say that K is universally complete if K = (K ⁵ ) \

Corollary 1. K ‘ = (j(K) ⁵ )\ i. e., (K ‘ ,j) is the universal completion of K.

An element 0 < g in an l-group G is singular if 0 C h C g implies h Л (g — h) = 0 for all h E G or, equivalently, if the order interval [0, x] of G is a Boolean algebra. An l-group G is a singular l-group if for each 0 < g E G, there exists a singular element s E G such that 0 < s C g. A singular f -ring is an f -ring whose underlying l-group is singular. Denote by C ^ (Q, F) a part of C ^ (Q, R) (see [1, p. 127] for definition) consisting of function with values in F, where F := Z _a , Z , R _a , R. The next two results can be derived making use of [3, Theorems 4.4.12 and 4.4.13] and [5, Theorem 3.5].

Theorem. Let K be an Archimedean f-ring and B = P(K) . There is a polar decomposition K ⁵ = G s ф G r ф H s ф H r where G s and G r are respectively a singular l-group and an erased vector lattice, both with zero multiplication, while H _s and H _r are respectively a singular f -ring and an f -algebra with erased scalar multiplication.

• <    Sketch of the Proof. The proof follows the same lines as the proof of [3, Theorem 3.5]. Put b s = [ K = Z Л ], b r = [ K = R a ] d s = [ K = Z ^л ] and d r = [ K = R ]. By Lemma 1 we observe that b _s ,b _r ,d _s ,d _r E B are pairwise disjoint and b s V b r V d _s V d r = 1. Arguing as in [4, Theorem 3.5] we define G s := b s Л Z a , H S := d s Л Z ^л , G r : = b r Л R a , and H := d r Л R . Now the f -rings G s := K ⁵ П G s ^ , G r := K ⁵ П G r ^ , H s := K ⁵ П H S ^ , and H r := K ⁵ П H r ^ are the desired polars of K ⁵ . >

  
  

Kusraev, A. G. and Tasoev, B. B.

Corollary 2. Let K be an Archimedean f -ring, B = P(K) , and Q the Stone representation space of B . Then there exist clopen sets Q k (k = 1, 2, 3,4) in Q such that Q = U k =i Q k and

K ^5X ^ C ^ (Qi, Z a ) Ф C ^ (Q 2 , Z) Ф C ^ (Q 3 , R a ) Ф C (Q4, R).