Co-rotation, co-rotation-annihilation, and involutive ordinal sum - - PowerPoint PPT Presentation

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Co-rotation, co-rotation-annihilation, and involutive ordinal sum - - PowerPoint PPT Presentation

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Sndor Jenei University of Pcs, Hungary Co-rotation, co-rotation-annihilation, and involutive ordinal sum constructions of residuated semigroups Supported by the SROP-4.2.2.C-11/1/ KONV-2012-0005 grant. 13. december 14., szombat

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Co-rotation, co-rotation-annihilation, and involutive ordinal sum constructions of residuated semigroups

Supported by the SROP-4.2.2.C-11/1/ KONV-2012-0005 grant.

Sándor Jenei

University of Pécs, Hungary

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Definitions

FLe-algebra = comm. RL + f , f is an arbitrary constant involutive = x’’= x, where x’ = x → f (observe f’=t) integral = t is its greatest element Group-like = involutive + f = t

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Conic representation: For any conic, involutive FLe-algebra

[S. Jenei, H. Ono, On Involutive FLe-monoids, Archive for Mathematical Logic, 51 (7-8), 719-738 (2012)]

Conic representation

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Twin Rotation

[S. Jenei, H. Ono, On Involutive FLe-monoids, Archive for Mathematical Logic, 51 (7-8), 719-738 (2012)]

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SLIDE 9 1 0.2 0.5 0.8 1 0.2 0.5 0.8 0.2 0.5 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1

An Unchartable Wilderness

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Group-like case

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0.2 0.4 0.6 0.8 1 0.5 1 0.6 0.4 0.2 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1

absorbent-continuous group-like FLe-algebras on subreal chains

[S. Jenei, F. Montagna, A classification of certain group-like FLe-chains, submitted]

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Call a chain ⟨X, ≤⟩ weakly real if X is order-dense and complete, there exists a dense Y ⊂X with |Y|<|X|, and for any x,y∈Y there exist u,v∈Y such that u>x,v>y, and there exists a strictly increasing function from [x,u] into [y,v]. An order dense chain is said to be subreal if its Dedekind-MacNeille completion is weakly real. Absorbent continuity = for x ∈ X-, a(x)⨂x = x, where a(x) = inf { u∈X- : u⨂x = x}

absorbent-continuous group-like FLe-algebras on subreal chains

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BL-algebras = divisibility (continuity) everywhere Absorbent continuity = continuity only at a few point of the domain of ⨂ (viewed as a two-place function)

absorbent-continuous group-like FLe-algebras on subreal chains

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Absorbent continuity = continuity only at a few point of the domain of ⨂ (viewed as a two-place function)

absorbent-continuous group-like FLe-algebras on subreal chains

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absorbent-continuous group-like FLe-algebras on subreal chains

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Absorbent continuity = continuity only at a few point of the domain of ⨂ (viewed as a two-place function)

absorbent-continuous group-like FLe-algebras on subreal chains

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1: Involutive ordinal sums

Theorem: The twin-rotation of the Clifford- style ordinal sum of any family of negative cones of group-like FLe-chains and their skew- duals is a group-like FLe-chain.

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Motivation

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Motivation

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Motivation

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Motivation

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Motivation

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Motivation

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Motivation

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Motivation

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2: Co-rotations

disconnected connected

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2: Co-rotations

disconnected connected

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Applications of the Rotation construction

Perfect and bipartite IMTL-algebras

[C. Noguera, F. Esteva, J. Gispert, Perfect and bipartite IMTL- algebras and disconnected rotations of basic semihoops, Archive for Mathematical Logic, 44 (2005), 869–886. ]

Free nilpotent minimum algebras

[M. Busaniche, Free nilpotent minimum algebras, Mathematical Logic Quartely 52 (3) (2006) 219–236. ]

Free Glivenko MTL-algebras

[R. Cignoli, A. Torrens, Free algebras in varieties of Glivenko MTL-algebras satisfying the equation 2(x2) = (2x)2, Studia Logica 83 (1-3) (2006) 157-181]

in the structural description of

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Applications of the Rotation construction

Nelson algebras

[M. Busaniche, R. Cignoli, Constructive Logic with Strong Negation as a Substructural Logic, Journal of Logic and Computation 20 (4) (2010) 761–793.]

in establishing a spectral duality for finitely generated nilpotent minimum algebras

[S. Aguzzoli, M. Busaniche, Spectral duality for finitely generated nilpotent minimum algebras, with applications, Journal of Logic and Computation 17 (4) (2007) 749–765.]

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3: Co-rotation-annihilations

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3: Co-rotation-annihilations

disconnected connected

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3: Co-rotation-annihilations

disconnected connected

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Thank you for Your attention!

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