Lattice-valued bornological vector spaces and systems Jan Paseka 1 - - PowerPoint PPT Presentation

Introduction Bornological vector spaces Bornological vector systems Future work References

Lattice-valued bornological vector spaces and systems

Jan Paseka1 Sergejs Solovjovs1 Milan Stehl´ ık2,3

1Masaryk University, Brno, Czech Republic

e-mail: paseka@math.muni.cz solovjovs@math.muni.cz

2Johannes Kepler University, Linz, Austria

e-mail: Milan.Stehlik@jku.at

3Universidad T´

ecnica Federico Santa Mar´ ıa, Valpara´ ıso, Chile

88th Workshop on General Algebra Warsaw University of Technology, Warsaw, Poland June 19 – 22, 2014

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 1/41

Introduction Bornological vector spaces Bornological vector systems Future work References

Acknowledgements

1 Jan Paseka and Sergejs Solovjovs were supported by ESF project

• No. CZ.1.07/2.3.00/20.0051 “Algebraic methods in Quantum

Logic” of Masaryk University in Brno, Czech Republic.

2 All the authors were supported by Aktion Project No. 67p5

(Austria – Czech Republic) “Algebraic, fuzzy and logical as- pects of statistical learning for cancer risk assessment”.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 2/41

Introduction Bornological vector spaces Bornological vector systems Future work References

Outline

1

Introduction

2

Lattice-valued bornological vector spaces

3

Lattice-valued bornological vector systems

4

Future work

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 3/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological spaces and their properties

L-bornological spaces

There exist well-known concepts of functional analysis, namely, bornological space and bounded map, which provide a conve- nient tool to study the notion of “boundedness”. The construct Born of bornological spaces and bounded maps has already found applications in Functional Analysis. In 2011, M. Abel and A. ˇ Sostak introduced the category L-Born

• f L-bornological spaces over a complete lattice L, in order to

start the development of the theory of lattice-valued bornology as an extension of the theory of crisp bornological spaces.

• M. Abel and A. ˇ

Sostak showed that L-Born is a topological construct provided that the lattice L is infinitely distributive.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 4/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological spaces and their properties

Topological properties of L-bornological spaces

• J. Paseka, S. Solovjovs, and M. Stehl´

ık gave the necessary and sufficient condition on the lattice L for the category L-Born to be topological (correcting an error of M. Abel and A. ˇ Sostak).

• J. Paseka, S. Solovjovs, and M. Stehl´

ık introduced the category L-Born of variable-basis lattice-valued bornological spaces (in the sense of S. E. Rodabaugh) over a subcategory L of the cat- egory Sup of -semilattices, showing the necessary and suffi- cient condition on L for the category L-Born to be topological. These topologicity conditions for the categories L-Born and L-Born make a striking difference with the case of the cat- egories for lattice-valued topology, which are topological pro- vided that their respective underlying lattices are complete.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 5/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological spaces and their properties

Topos-theoretic properties of L-bornological spaces

• J. Paseka, S. Solovjovs, and M. Stehl´

ık showed that for a frame L (satisfying one additional condition), the full subcategory L-Borns of L-Born of strict L-bornological spaces (in the sense

• f M. Abel and A. ˇ

Sostak) is a topological construct, which is a well-fibred quasitopos, i.e., provides a topological universe. This result makes another striking difference with the cate- gories of lattice-valued topological spaces, which fail both to be cartesian closed and to have representable extremal partial morphisms, and which, therefore, do not provide quasitopoi.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 6/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological vector spaces and their properties

Lattice-valued bornological vector spaces

Motivated by the theory of bornological vector spaces, this talk introduces the category L-VBorn of L-bornological vector spaces over a complete lattice L. We show that for some complete lattices L, L-VBorn is topo- logical over the category Vec of vector spaces, coming to the conclusion that L-VBorn is a topologically algebraic construct. The category L-VBorn and its respective results are additionally extended to variable-basis (in the sense of S. E. Rodabaugh)

• ver subcategories of the category Sup of -semilattices.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 7/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological vector systems

Lattice-valued bornological vector systems

Stimulated by the concept of topological system of S. Vickers (a common setting for both point-set and point-free topologies),

• J. Paseka, S. Solovjovs, and M. Stehl´

ık introduced the category L-BornSys of L-bornological systems, and showed that the cat- egory L-Born is isomorphic to its full reflective subcategory. This talk introduces the category L-VBornSys of L-bornological vector systems, and shows that the category L-VBorn is isomor- phic to a full reflective subcategory of the category L-VBornSys. The category L-VBornSys and its respective results are ex- tended to variable-basis over subcategories of the category Sup.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 8/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological vector systems

Lattice-valued bornology and cancer research

The purpose of our investigation is the development of a lattice- valued analogue of the Hausdorff dimension in convex bornolog- ical spaces of J. Almeida and L. Barreira, to provide a new technique for the study of cancer-related diseases in humans. Our machinery will take into account the presence of fuzziness in the real world phenomena (in the sense of L. A. Zadeh).

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 9/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological construct of L-bornological spaces

Bornological spaces and bounded maps

Every map X

f

− → Y gives rise to the forward powerset operator PX

f →

− − → PY , which is defined by f →(S) = {f (s) | s ∈ S}. Definition 1 A bornological space is a pair (X, B), where X is a set, and B (a bornology on X) is a subfamily of PX (the elements of which are called bounded sets), which satisfy the following axioms:

1 X = B(=

B∈B B);

2 if B ∈ B and D ⊆ B, then D ∈ B; 3 if S ⊆ B is finite, then S ∈ B.

Given bornological spaces (X1, B1), (X2, B2), a map X1

f

− → X2 is bounded provided that f →(B1) ∈ B2 for every B1 ∈ B1. Born is the construct of bornological spaces and bounded maps.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 10/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological construct of L-bornological spaces

L-bornological spaces and L-bounded maps

Given a complete lattice L, every map X

f

− → Y provides the forward L-powerset operator LX

f →

L

− − → LY with (f →

L (B))(y) = f (x)=y B(x).

Definition 2 (M. Abel and A. ˇ Sostak) An L-bornological space is a pair (X, B), where X is a set, and B (an L-bornology on X) is a subfamily of LX (the elements of which are called bounded L-sets), which satisfy the following axioms:

1

B∈B B(x) = ⊤L for every x ∈ X;

2 if B ∈ B and D B, then D ∈ B; 3 if S ⊆ B is finite, then S ∈ B.

Given L-bornological spaces (X1, B1), (X2, B2), a map X1

f

− → X2 is L-bounded provided that f →

L (B1) ∈ B2 for every B1 ∈ B1. L-Born

is the construct of L-bornological spaces and L-bounded maps.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 11/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological construct of L-bornological spaces

Crisp bornological spaces as L-bornological spaces

Remark 3 The construct 2-Born is concretely isomorphic to the construct Born, where 2 = {⊥, ⊤} is the two-element complete lattice. Every subset B of a set X can be considered as an L-characteristic map X

BL

− → L, which is defined by BL(x) =

• ⊤L,

x ∈ B ⊥L,

• therwise.

Proposition 4 Given a complete lattice L with at least two elements, there ex- ists a full concrete embedding 2-Born

E

L-Born defined by

E(X, B) = (X, BL) with BL = {C ∈ LX | C BL for some B ∈ B}.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 12/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological construct of L-bornological spaces

Ideal complete distributivity

Definition 5 Given a complete lattice L, a subset S ⊆ L is called a lattice ideal

• f L provided that

1 if a ∈ L and a b for some b ∈ S, then a ∈ S; 2 if T ⊆ S is finite, then T ∈ S.

Definition 6 A complete lattice L is called ideally completely distributive at ⊤L provided that for every non-empty family {Si | i ∈ I} of lattice ideals

• f L,

i∈I( Si) = ⊤L implies h∈H( i∈I h(i)) = ⊤L, where H is

the set of choice functions on

i∈I Si, which are maps I h

− →

i∈I Si

such that h(i) ∈ Si for every i ∈ I.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 13/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological construct of L-bornological spaces

L-Born is a topological construct

Theorem 7 The construct (L-Born, |−|) is topological iff L is ideally completely distributive at ⊤L. By the above theorem, the constructs 2-Born (crisp approach) and [0, 1]-Born (fuzzy approach) are topological. From now on, let L be ideally completely distributive at ⊤L.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 14/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topologically algebraic construct of L-bornological vector spaces

Products of L-bornological spaces

Given a set-indexed family of maps (Xi

Bi

− → L)i∈I, define a map

• i∈I Xi

⊗i∈I Bi

− − − − → L by the formula (⊗i∈IBi)((xi)i∈I) =

i∈I Bi(xi).

Proposition 8 The product of a family ((Xi, Bi))i∈I of L-bornological spaces is given by the source ((

i∈I Xi, B⊗) πj

− → (Xj, Bj))j∈I, where

i∈I Xi πj

− → Xj is the j-th projection map and B⊗ = {B ∈ LX | B ⊗i∈IBi such that Bi ∈ Bi for every i ∈ I}.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 15/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topologically algebraic construct of L-bornological vector spaces

L-bornological vector spaces

Definition 9 An L-bornological vector space is a tuple (X, +, ∗, B), where (X, +, ∗) is a vector space over the field of real numbers R, and (X, B) is an L-bornological space such that:

1 the map (X, B) × (X, B) +

− → (X, B) is L-bounded;

2 the map (R, BRL) × (X, B) ∗

− → (X, B) is L-bounded, where BRL is the image under the embedding E of Proposition 4 of the bornology BR of bounded sets (in the classical sense) on R. Given L-bornological vector spaces (X1, +, ∗, B1), (X2, +, ∗, B2), a map X1

f

− → X2 is L-bounded linear provided that f is L-bounded and linear (i.e., f (x + y) = f (x) + f (y) and f (r ∗ x) = r ∗ f (x) for every x, y ∈ X1 and every r ∈ R). L-VBorn is the construct of L-bornological vector spaces and L-bounded linear maps.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 16/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topologically algebraic construct of L-bornological vector spaces

L-VBorn is topologically algebraic

Vec is the category of vector spaces and linear maps. Theorem 10 The concrete category (L-VBorn, | − |) over Vec is topological. Theorem 11 The construct (L-VBorn, | − |) is topologically algebraic. Proof. It is enough to show the following two properties.

1 The forgetful functor L-VBorn

|−|

− − → Set has a left adjoint.

2 The category L-VBorn is (Epi, Initial Source)-factorizable. Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 17/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topologically algebraic construct of L-bornological vector spaces

L-convex L-bornological vector spaces

Definition 12 Given a vector space (X, +, ∗) and an L-set A ∈ LX, A is called

1 L-circled provided that for every x ∈ X and every r ∈ R such

that |r| 1, A(x) A(r ∗ x);

2 L-convex provided that for every x1, x2 ∈ X and every positive

r1, r2 ∈ R with r1 +r2 = 1, A(x1)∧A(x2) A(r1 ∗x1 +r2 ∗x2);

3 L-disk provided that A is both L-convex and L-circled.

An L-disked hull Adh of A is defined by Adh = {B ∈ LX | B is an L-disk and A B}. The disked hull Adh of A is an L-disk. Definition 13 An L-bornological vector space (X, +, ∗, B) is called L-convex pro- vided that Bdh ∈ B for every B ∈ B. L-CVBorn is the full subcate- gory of L-VBorn of L-convex L-bornological vector spaces.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 18/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological category of variable-basis lattice-valued bornological spaces

Variable-basis lattice-valued forward powerset operator

Set is the category of sets and maps. Sup is the category of -semilattices and -preserving maps. Proposition 14 Every subcategory L of Sup provides a functor Set × L

(−)→

− − − → Sup, which is defined by ((X1, L1)

(f ,ψ)

− − − → (X2, L2))→ = LX1

1 (f ,ψ)→

− − − − → LX2

2 ,

where ((f , ψ)→(B))(x2) =

f (x1)=x2 ψ ◦ B(x1).

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 19/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological category of variable-basis lattice-valued bornological spaces

Variable-basis lattice-valued bornology

Definition 15 Given a subcategory L of Sup, L-Born is the category, which is concrete over the product category Set × L, whose

• bjects are triples (X, L, B), where L is an L-object, and (X, B) is

an L-bornological space; and whose morphisms (X1, L1, B1)

(f ,ψ)

− − − → (X2, L2, B2) (called L-bounded maps) consist of a map X1

f

− → X2 and an L-morphism L1

ψ

− → L2 such that (f , ψ)→(B1) ∈ B2 for every B1 ∈ B1. Example 16 Every complete lattice L is a -semilattice, which provides a subcat- egory L of Sup, whose only morphism is the identity map L

1L

− → L. The categories L-Born and L-Born are then concretely isomorphic.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 20/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological category of variable-basis lattice-valued bornological spaces

A particular category of -semilattices

Given a -semilattice homomorphism L1

ψ

− → L2, there exists a - preserving map L2

ψ⊢

− − → L1 with ψ⊢(b) = {a ∈ L1 | ψ(a) b}. Definition 17 L⊢ is the subcategory of Sup, whose objects L are ideally completely distributive at ⊤L, and whose morphisms L1

ψ

− → L2 are such that the map L2

ψ⊢

− − → L1 has the following property: ψ⊢( S) =

s∈S ψ⊢(s) for every S ⊆ L2 such that S = ⊤L2.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 21/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological category of variable-basis lattice-valued bornological spaces

L-Born is a topological category

Theorem 18 The concrete category (L-Born, |−|) is topological over the product category Set × L iff L is a subcategory of L⊢. Example 19 An example of L, which gives a topological category L-Born, is the subcategory of Sup, whose objects satisfy the condition of the category L⊢, and whose morphisms are -semilattice isomorphisms.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 22/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological category of variable-basis lattice-valued bornological spaces

Variable-basis lattice-valued bornological vector spaces

Definition 20 Given a subcategory L of Sup, L-VBorn is the category, which is concrete over the product category Vec × L, whose

• bjects (L-bornological vector spaces) are L-bornological vector

spaces (X, +, ∗, L, B) such that L is an L-object; and whose morphisms (L-bounded linear maps) (X1, +, ∗, L1, B1)

(f ,ψ)

− − − → (X2, +, ∗, L2, B2) are L-bounded maps (X1, L1, B1)

(f ,ψ)

− − − → (X2, L2, B2) such that the map (X1, +, ∗) f − → (X2, +, ∗) is linear. From now on, let L be a subcategory of the category L⊢.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 23/41

Introduction Bornological vector spaces Bornological vector systems Future work References Topological category of variable-basis lattice-valued bornological spaces

L-VBorn is topologically algebraic over Set × L

Theorem 21 The concrete category (L-VBorn, | − |) over Vec × L is topological. Theorem 22 The concrete category (L-VBorn, |−|) over Set×L is topologically algebraic.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 24/41

Introduction Bornological vector spaces Bornological vector systems Future work References L-bornological vector systems

be-lattices and their homomorphisms

Definition 23 A be-lattice is a poset C, which has finite and non-empty . Given be-lattices C1 and C2, a be-lattice homomorphism is a map C1

ϕ

− → C2, which preserves finite . be-Lat stands for the construct of be-lattices and be-lattice homomorphisms.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 25/41

Introduction Bornological vector spaces Bornological vector systems Future work References L-bornological vector systems

L-bornological systems

Definition 24 An L-bornological system is a triple (X, C, | =), where X is a set, C is a be-lattice, and X × C

| =

− → L is a map (L-satisfaction relation on (X, C)), which satisfies the following axioms:

1 for every x ∈ X,

c∈C |

=(x, c) = ⊤L;

2 if c ∈ C and D ∈ LX are such that D |

=(−, c), then there exists c′ ∈ C such that D = | =(−, c′);

3 for every x ∈ X, the map C

| =(x,−)

− − − − → L preserves finite .

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 26/41

Introduction Bornological vector spaces Bornological vector systems Future work References L-bornological vector systems

L-bornological system morphisms

Definition 25 Given L-bornological systems (X1, C1, | =1) and (X2, C2, | =2), an L-bornological system morphism (also called L-bounded map) (X1, C1, | =1)

(f ,ϕ)

− − − → (X2, C2, | =2) consists of a map X1

f

− → X2 and a be-lattice homomorphism C1

ϕ

− → C2 such that for every x2 ∈ X2 and every c1 ∈ C1, | =2(x2, ϕ(c1)) =

f (x1)=x2 |

=1(x1, c1). Definition 26 L-BornSys is the category of L-bornological systems and L-bounded maps, which is concrete over the product category Set × be-Lat.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 27/41

Introduction Bornological vector spaces Bornological vector systems Future work References L-bornological vector systems

Spaces versus systems

Theorem 27

1 There exists a full embedding

L-Born

E

L-BornSys,

which is defined by E((X1, B1)

f

− → (X2, B2)) = (X1, B1, | =1)

(f ,¯ f →

L )

− − − − → (X2, B2, | =2), where | =i(x, B) = B(x), and ¯ f →

L

is the restriction of f →

L

to B1, B2.

2 There exists a functor L-BornSys

Spat

− − − → L-Born given by Spat((X1, C1, | =1)

(f ,ϕ)

− − − → (X2, C2, | =2)) = (X1, {Ext1(c) | c ∈ C1}) f − → (X2, {Ext2(c) | c ∈ C2}) with (Exti(c))(x) = | =i(x, c).

3 Spat is a left-adjoint-left-inverse to E. Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 28/41

Introduction Bornological vector spaces Bornological vector systems Future work References L-bornological vector systems

L-bornological vector systems

Definition 28 An L-bornological vector system is a quintuple (X, +, ∗, C, | =), where (X, +, ∗) is a vector space over R, and (X, C, | =) is an L- bornological system, which satisfy the next axioms:

1 for every c1, c2 ∈ C, there exists c ∈ C with |

=(x1, c1) ∧ | =(x2, c2) | =(x1 + x2, c) for every x1, x2 ∈ X;

2 for every S ∈ BR, and every c ∈ C, there exists c′ ∈ C such

that {| =(x′, c) | s ∗ x′ = x for some s ∈ S} | =(x, c′) for every x ∈ X.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 29/41

Introduction Bornological vector spaces Bornological vector systems Future work References L-bornological vector systems

L-bornological vector system morphisms

Definition 29 Given L-bornological vector systems (X1, +, ∗, C1, | =1) and (X2, +, ∗, C2, | =2), an L-bornological vector system morphism (also called L-bounded linear map) (X1, +, ∗, C1, | =1)

(f ,ϕ)

− − − → (X2, +, ∗, C2, | =2) is an L-bounded map (X1, C1, | =1)

(f ,ϕ)

− − − → (X2, C2, | =2) such that (X1, +, ∗) f − → (X2, +, ∗) is a linear map. Definition 30 L-VBornSys is the category of L-bornological vector systems and L-bounded linear maps, concrete over the category Vec × be-Lat.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 30/41

Introduction Bornological vector spaces Bornological vector systems Future work References L-bornological vector systems

Spaces versus systems

Theorem 31

1 There exists a full embedding L-VBorn

E

L-VBornSys,

which is defined by E((X1, +, ∗, B1)

f

− → (X2, +, ∗, B2)) = (X1, +, ∗, B1, | =1)

(f ,¯ f →

L )

− − − − → (X2, +, ∗, B2, | =2), with | =i(x, B) = B(x), and ¯ f →

L

the restriction of f →

L

to B1, B2.

2 There exists a functor L-VBornSys

Spat

− − − → L-VBorn given by Spat((X1, +, ∗, C1, | =1)

(f ,ϕ)

− − − → (X2, +, ∗, C2, | =2)) = (X1, +, ∗, {Ext1(c) | c ∈ C1}) f − → (X2, +, ∗, {Ext2(c) | c ∈ C2}) with (Exti(c))(x) = | =i(x, c).

3 Spat is a left-adjoint-left-inverse to E. Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 31/41

Introduction Bornological vector spaces Bornological vector systems Future work References L-bornological vector systems

L-convex L-bornological vector systems

Definition 32 An L-bornological vector system (X, +, ∗, C, | =) is called L-convex provided that for very c ∈ C, there exists c′ ∈ C such that (Ext(c))dh = Ext(c′). L-CVBornSys is the full subcategory of L-VBornSys of L-convex L-bornological vector systems. Theorem 33

1 There exists the restriction L-CVBorn CE L-CVBornSys

• f the full embedding L-VBorn

E

L-VBornSys .

2 There exists the restriction L-CVBornSys

CSpat

− − − → L-CVBorn of the functor L-VBornSys

Spat

− − − → L-VBorn.

3 CSpat is a left-adjoint-left-inverse to CE. Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 32/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological vector systems

Variable-basis lattice-valued bornological systems

Definition 34 Given a subcategory L of Sup, L-BornSys is the category, concrete

• ver the product category Set × L × be-Lat, whose
• bjects are tuples (X, L, C, |

=), where L is an L-object, and (X, C, | =) is an L-bornological system; and whose morphisms (X1, L1, C1, | =1)

(f ,ψ,ϕ)

− − − − → (X2, L2, C2, | =2) (also called L- bounded maps) have a map X1

f

− → X2, an L-morphism L1

ψ

− → L2, and a be-lattice homomorphism C1

ϕ

− → C2 such that for every x2 ∈ X2 and every c1 ∈ C1, | =2(x2, ϕ(c1)) =

f (x1)=x2 ψ ◦ |

=1(x1, c1).

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 33/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological vector systems

Spaces versus systems

Theorem 35

1 There exists a full embedding

L-Born

E

L-BornSys

given by E((X1, L1, B1)

(f ,ψ)

− − − → (X2, L2, B2)) = (X1, L1, B1, | =1)

(f ,ψ,(f ,ψ)

→)

− − − − − − − − → (X2, L2, B2, | =2), | =i(x, B) = B(x), and (f , ψ)

→ is the restriction of (f , ψ)→ to B1, B2.

2 There exists a functor L-BornSys

Spat

− − − → L-Born given by Spat((X1, L1, C1, | =1)

(f ,ψ,ϕ)

− − − − → (X2, L2, C2, | =2)) = (X1, L1, {Ext1(c) | c ∈ C1})

(f ,ψ)

− − − → (X2, L2, {Ext2(c) | c ∈ C2}), where (Exti(c))(x) = | =i(x, c).

3 Spat is a left-adjoint-left-inverse to E. Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 34/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological vector systems

Variable-basis lattice-valued bornological vector systems

Definition 36 Given a subcategory L of Sup, L-VBornSys is the category, concrete

• ver the product category Vec × L × be-Lat, whose
• bjects (L-bornological vector systems) are L-bornological vector

systems (X, +, ∗, L, C, | =) such that L is an L-object; and whose morphisms (L-bounded linear maps) (X1, +, ∗, L1, C1, | =1)

(f ,ψ,ϕ)

− − − − → (X2, +, ∗, L2, C2, | =2) are L-bounded maps (X1, L1, B1)

(f ,ψ,ϕ)

− − − − → (X2, L2, B2) such that the map X1

f

− → X2 is linear.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 35/41

Introduction Bornological vector spaces Bornological vector systems Future work References Lattice-valued bornological vector systems

Spaces versus systems

Theorem 37

1 There exists a full embedding L-VBorn

E

L-VBornSys

defined by E((X1, +, ∗, L1, B1)

(f ,ψ)

− − − → (X2, +, ∗, L2, B2)) = (X1, +, ∗, L1, B1, | =1)

(f ,ψ,(f ,ψ)

→)

− − − − − − − − → (X2, +, ∗, L2, B2, | =2), | =i(x, B)=B(x), (f , ψ)

→is the restriction of (f , ψ)→ to B1, B2.

2 There

exists a functor L-BornSys

Spat

− − − → L-Born de- fined by Spat((X1, +, ∗, L1, C1, | =1)

(f ,ψ,ϕ)

− − − − → (X2, +, ∗, L2, C2, | =2)) = (X1, +, ∗, L1, {Ext1(c) | c ∈ C1})

(f ,ψ)

− − − → (X2, +, ∗, L2, {Ext2(c) | c ∈ C2}), (Exti(c))(x) = | =i(x, c).

3 Spat is a left-adjoint-left-inverse to the embedding E. Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 36/41

Introduction Bornological vector spaces Bornological vector systems Future work References Final remarks

Conclusion

Following the idea of L-bornological space over a complete lat- tice L of M. Abel and A. ˇ Sostak, this talk has introduced the category L-VBorn of L-bornological vector spaces, which ap- peared to be a topologically algebraic construct. We have also introduced a variable-basis modification L-VBorn

• f L-VBorn (in the sense of S. E. Rodabaugh) over a subcate-

gory L of the category Sup of -semilattices, and showed that L-VBorn is topologically algebraic over the category Set × L. Moreover, we have introduced the category L-VBornSys of L-bornological vector systems, and showed that the category L-VBorn is isomorphic to its full reflective subcategory. This result has been additionally extended variable-basis.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 37/41

Introduction Bornological vector spaces Bornological vector systems Future work References Open problems

Open problems

Problem 38 What will be a lattice-valued (possibly, variable-basis) analogue of the crisp “topology – bornology” dualities? Problem 39 What will be the analogue of lattice-valued (possibly, variable-basis) bornological vector spaces, which is based in lattice-valued vector spaces in the sense of lattice-valued algebra? Problem 40 What is the nature of the category L-VBornSys of lattice-valued bornological vector systems, namely, whether it is essentially alge- braic over its ground category?

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 38/41

Introduction Bornological vector spaces Bornological vector systems Future work References References

References I

• M. Abel and A. ˇ

Sostak, Towards the theory of L-bornological spaces,

• Iran. J. Fuzzy Syst. 8 (2011), no. 1, 19–28.
• J. Ad´

amek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: The Joy of Cats, Dover Publications (Mineola, New York), 2009.

• A. Di Nola and G. Gerla, Lattice valued algebras, Stochastica 11

(1987), no. 2-3, 137–150.

• G. Gierz, K. Hofmann, K. Keimel, J. Lawson, M. Mislove, and
• D. S. Scott, Continuous Lattices and Domains, Cambridge Univer-

sity Press, 2003.

• H. Hogbe-Nlend, Bornologies and Functional Analysis, Mathematics

Studies, vol. 26, North-Holland Publishing Company, 1977.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 39/41

Introduction Bornological vector spaces Bornological vector systems Future work References References

References II

• J. Paseka, S. Solovyov, and M. Stehl´

ık, Lattice-valued bornological systems, submitted to Fuzzy Sets Syst. (Special Issue: LINZ 2013).

• J. Paseka, S. Solovyov, and M. Stehl´

ık, On a topological universe of L-bornological spaces, submitted to Soft Comput.

• J. Paseka, S. Solovyov, and M. Stehl´

ık, On the category of lattice- valued bornological vectors spaces, J. Math. Anal. Appl. 419 (2014),

• no. 1, 138–155.
• S. E. Rodabaugh, Categorical Foundations of Variable-Basis Fuzzy

Topology, Mathematics of Fuzzy Sets: Logic, Topology and Mea- sure Theory (U. H¨

• hle and S. E. Rodabaugh, eds.), Kluwer Academic

Publishers, 1999, pp. 273–388.

Lattice-valued bornological vector spaces and systems Sergejs Solovjovs Masaryk University 40/41

Introduction Bornological vector spaces Bornological vector systems Future work References

Thank you for your attention!

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