Tue, 15 Apr 2025, 1:25 pm MDT
This study is based on ‘algebraic-analytic’ approach to some non-associative hyper-structures with application to physical systems. This was with a view of understanding the non-associatve algebraic behaviour of: (i) the elementary particle physics (lepton group) which forms an \(H_v\)-group (Nezhad et al. 2012), the elementary particles (including the Higgs Boson) originally studied by (Davvaz et al. 2020) and (ii) some dismutation reactions in chemical systems (Davvaz et al. 2012). Non-commutative groupoid (quasigroup and loop) was used to construct polygroupoid (polyquasigroup, polyloop) and examples given. The Kuratowski closure axioms was used for an appropriate closure operator that is nuclear in nature (relative to polygroupoid) to produce a Kuratowski induced topological space and consequently a polygroupoid-topological space. This study introduced and investigated the properties of left (right) nuclei Kuratowski closure operator induced topological space on polygroupoid (polyquasigroup, polyloop). This was used to analyse (with the aid of probability) the nuclear and alternative properties of the lepton group. The analysis of algebraic properties (with the aid of probability of elements) in dismutation reaction of some chemical systems of Tin (\(Sn\)), Indium (\(In\)) and Vanadium (\(V\)) which are represented by hyper-algebraic structures were carried out. For dismutation reaction, it was discovered that the \(N_\lambda\)-probability, \(N_\mu\)-probability and \(N_\rho\)-probability for each of \((S_{Sn},\oplus)\), \((S_{In},\oplus)\) and \((S_V,\oplus)\) are all less than 1.000. This implies that, \((S_{Sn},\oplus)\), \((S_{In},\oplus)\) and \((S_V,\oplus)\) and are non-associative hyper-algebraic structures. Also, from the result obtained for FLEX-probability, it can be concluded that, \((S_{Sn},\oplus)\), \((S_{In},\oplus)\) and \((S_V,\oplus)\) have flexible elements because the value of FLEX probability is 1.000. Further more, none of \((S_{Sn},\oplus)\), \((S_{In},\oplus)\) and \((S_V,\oplus)\) satisfied the left alternative and right alternative properties because the value of LAP-probability and RAP-probability are less than 1.000.
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