{'fraction_non_alphanumeric': 0.06164383561643835, 'fraction_numerical': 0.009784735812133072, 'mean_word_length': 4.300518134715026, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
States with values in the Łukasiewicz groupoid

Abstract In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let ∗ stand for the Łukasiewicz t-norm on [0, 1]2. Let us consider the operation ⋄ on [0, 1] by setting x ⋄ y = (x⊥ ∗y⊥)⊥ ∗ (x∗y)⊥, where x⊥ = 1−x. Let us call the triple L = ([0, 1], ⋄, 1) the Łukasiewicz groupoid. Let B be a Boolean algebra. Denote by L(B) the set of all L-valued measures (L-valued states). We show as a main result of this paper that the family L(B) consists precisely of the union of classical real states and Z2-valued states. With the help of this result we characterize the L-valued states on orthomodular posets. Since the orthomodular posets are often understood as “quantum logics” in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L-valued state.