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Keywords:
effect-tribe; tribe; monotone $\sigma $-complete effect algebra; Riesz decomposition property; MV-algebra
effect-tribe; tribe; monotone $\sigma $-complete effect algebra; Riesz decomposition property; MV-algebra
Summary:
We show that an effect tribe of fuzzy sets ${\mathcal T}\subseteq [0,1]^X$ with the property that every $f\in {\mathcal T}$ is ${\mathcal B}_0({\mathcal T})$-measurable, where ${\mathcal B}_0({\mathcal T})$ is the family of subsets of $X$ whose characteristic functions are central elements in ${\mathcal T}$, is a tribe. Moreover, a monotone $\sigma$-complete effect algebra with RDP with a Loomis-Sikorski representation $(X, {\mathcal T},h)$, where the tribe ${\mathcal T}$ has the property that every $f\in {\mathcal T}$ is ${\mathcal B}_0({\mathcal T})$-measurable, is a $\sigma$-MV-algebra.
References:
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