Strongly Recurrent Transformation Groups

We define the notions of strong and strict recurrency for actions of countable ordered groups on $\sigma$-finite non atomic measure spaces with quasi-invariant measures. We show that strong recurrency is equivalent to non existence of weakly wandering sets of positive measure. We also show that for certain p.m.p ergodic actions the system is not strictly recurrent, which shows that strong and strict recurrency are not equivalent.

Cevian properties in ideal lattices of Abelian ℓ-groups

We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

Preliminary Study: Immediate Effect of the Slow Deep Breathing Exercise and Sustained Maximal Inspiration with Volume-Oriented Incentive Spirometry on the Diffusing Lung Capacity in Healthy Young Participants

Slow-deep breathing exercise (SDBE) while using a volume-oriented type of incentive spirometry (VIS) device (SDBE/VIS) is one of the techniques in chest physical therapy designed to improve lung volume and oxygenation. However, the immediate effect of SDBE/VIS paired with a sustained maximal inspiration (SMI) on diffusing lung capacity (DLC), has not been documented. This preliminary study aimed to evaluate the immediate effect of SDBE/VIS paired with the SMI technique on DLC in healthy participants. Twenty healthy sedentary adults (11 males and 9 females) aged 20 to 23 years were recruited into this cross-over study and randomly assigned to one of 2 ordered groups; SDBE/VIS paired with SMI followed by SDBE/VIS alone or SDBE/VIS alone followed by SDBE/VIS with SMI. During each of the 2-testing sessions, the participants performed 3 sets of 5 SDBE/VIS trials which were controlled by ensuring that each participant reached their maximal vital capacity; derived from their pulmonary function test. The DLC for carbon monoxide (DLCO), alveolar volume (VA), and total lung capacity (TLC) was assessed prior to (baseline) and after each of the 5 experimental trials. A 3-min rest period was given between each set of 5 trials and 1 week between testing sessions. The results demonstrated that SDBE/VIS paired with the SMI technique significantly increased both the DLCO (mL/min/mmHg and %) and VA (L and %) when compared to the baseline values. Whereas, SDBE/VIS without using the SMI technique did not alter the DLCO, VA or the DLCO/VA. However, both SDBE/VIS with and without the SMI technique showed a significant increase the TLC (L and %), with the TLC from SDBE/VIS paired with SMI was significantly higher than when the SDBE /VIS was performed without using the SMI technique. Thus, it appears that performing SDBE/VIS exercised paired with the SMI technique has a significant and positive effect on DLC.

Sums and products of intervals in ordered groups and fields

We show that the sum of two intervals in an ordered dense Abelian group is also an interval such that the endpoints of the sum are equal to the sums of the endpoints. We prove analogous statements concerning to the product of two intervals.

A relationship between the category of chain MV-algebras and a subcategory of abelian groups

The category of MV-algebras is equivalent to the category of abelian lattice ordered groups with strong units. In this article we introduce the category of circled abelian groups and prove that the category of chain MV-algebras is isomorphic with the category of chain circled abelian groups. In the last section we show that the category of chain MV-algebras is a subcategory of abelian cyclically ordered groups.

Equivalence à la Mundici for commutative lattice-ordered monoids

We provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation $$x \mapsto -x$$ x ↦ - x . The primitive operations are $$+$$ + , $$\vee $$ ∨ , $$\wedge $$ ∧ , 0, 1, $$-1$$ - 1 . A prime example of these structures is $$\mathbb {R}$$ R , with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation $$x \mapsto \lnot x$$ x ↦ ¬ x . The primitive operations are $$\oplus $$ ⊕ , $$\odot $$ ⊙ , $$\vee $$ ∨ , $$\wedge $$ ∧ , 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra $$[0, 1]\subseteq \mathbb {R}$$ [ 0 , 1 ] ⊆ R . We obtain the original Mundici’s equivalence as a corollary of our main result.