Relatively additive states on quantum logics

@article{Pták2005,
abstract = {In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too.},
author = {Pták, Pavel, Weber, Hans},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {(weak) state on quantum logic; Greechie paste job; Boolean algebra; quantum logic; Boolean algebra; weak state},
language = {eng},
number = {2},
pages = {327-338},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Relatively additive states on quantum logics},
url = {http://eudml.org/doc/249531},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Pták, Pavel
AU - Weber, Hans
TI - Relatively additive states on quantum logics
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 2
SP - 327
EP - 338
AB - In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for $2$-additive three-valued states, and in case of Greechie lattices we show that one can even construct many $2$-additive two-valued states. Some open questions are posed, too.
LA - eng
KW - (weak) state on quantum logic; Greechie paste job; Boolean algebra; quantum logic; Boolean algebra; weak state
UR - http://eudml.org/doc/249531
ER -