ORIGINAL_ARTICLE
Everyday physics of extended bodies or why functionals need analyzing
Functionals were discovered and used by Volterra over a century ago in his study of the motions of viscous elastic materials and electromagnetic fields. The need to precisely account for the qualitative effects of the cohesion and shape of the domains of these functionals was the major impetus to the development of the branch of mathematics known as topology, and today large numbers of mathematicians still devote their work to a detailed technical analysis of functionals. Yet the concept needs to be understood by all people who want to fully participate in 21st century society. Through some explicit use of mathematical categories and their transformations, functionals can be treated in a way which is non-technical and yet permits considerable reliable development of thought. We show how a deformable body such as a storm cloud can be viewed as a kind of space in its own right, as can an interval of time such as an afternoon; the infinite-dimensional spaces of configurations of the body and of its states of motion are constructed, and the role of the infinitesimal law of its motion revealed. We take nilpotent infinitesimals as given, and follow Euler in defining real numbers as ratios of infinitesimals.
http://cgasa.sbu.ac.ir/article_40434_9767cf7e5bb1b3e6e8b6cb442625ee0f.pdf
2017-01-01T11:23:20
2019-10-21T11:23:20
9
19
Functionals
physics
F. William
Lawvere
wlawvere@buffalo.edu
true
1
Professor Emeritus, Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.
Professor Emeritus, Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.
Professor Emeritus, Department of Mathematics, University at Buffalo, Buffalo, New York 14260-2900, United States of America.
AUTHOR
[1] Bell, J.L., A Primer of Infinitesimal Analysis", Cambridge University Press, Cambridge,1998.
1
[2] Kock, A., Synthetic Differential Geometry", Cambridge University Press (2nd Edition),Cambridge, 2006.
2
[3] Lavendhomme, R., Basic Concepts of Synthetic Differential Geometry", KluwerAcademic Publishers Dordrecht, Kluwer Texts in Mathematical Sciences 13, 1996.
3
[4] Lawvere, F.W., Toward the description in a smooth topos of the dynamically possiblemotions and deformations of a continuous body, Cah. Topol. Geom. Differ. Categ.XXI (1980), 337-392.
4
[5] Lawvere, F.W., Comments on the development of topos theory, in the book:Development of Mathematics 1950-2000", J.P. Pier (Ed) Birkhauser Verlag, Basel,(2000), 715-734. See also in TAC Reprints 24 (2014), 1-22 (with author commentary).
5
[6] Moerdijk, I. and Reyes, G., Models for Smooth Infinitesimal Analysis", Springer-Verlag, New York, 1991.
6
ORIGINAL_ARTICLE
Localic maps constructed from open and closed parts
Assembling a localic map $f\colon L\to M$ from localic maps $f_i\colon S_i\to M$, $i\in J$, defined on closed resp. open sublocales $(J$ finite in the closed case$)$ follows the same rules as in the classical case. The corresponding classical facts immediately follow from the behavior of preimages but for obvious reasons such a proof cannot be imitated in the point-free context. Instead, we present simple proofs based on categorical reasoning. There are some related aspects of localic preimages that are of interest, though. They are investigated in the second half of the paper.
http://cgasa.sbu.ac.ir/article_15806_f90dc6ec251a402f3ff01305864296bd.pdf
2017-01-01T11:23:20
2019-10-21T11:23:20
21
35
frame
locale
sublocale
sublocale lattice
open sublocale
closed sublocale
localic map
preimage
Boolean frame
linear frame
Ales
Pultr
pultr@kam.ms.mff.cuni.cz
true
1
Department of Applied Mathematics and ITI, MFF, Charles University, Malostransk'e n'am. 24, 11800 Praha 1, Czech Republic.
Department of Applied Mathematics and ITI, MFF, Charles University, Malostransk'e n'am. 24, 11800 Praha 1, Czech Republic.
Department of Applied Mathematics and ITI, MFF, Charles University, Malostransk'e n'am. 24, 11800 Praha 1, Czech Republic.
AUTHOR
Jorge
Picado
picado@mat.uc.pt
true
2
CMUC, Department of Mathematics, University of Coimbra, Apar\-ta\-do 3008, 3001-501 Coimbra, Portugal.
CMUC, Department of Mathematics, University of Coimbra, Apar\-ta\-do 3008, 3001-501 Coimbra, Portugal.
CMUC, Department of Mathematics, University of Coimbra, Apar\-ta\-do 3008, 3001-501 Coimbra, Portugal.
AUTHOR
[1] J. R. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972), 5-32.
1
[2] P. T. Johnstone, "Stone Spaces", Cambridge Univ. Press, Cambridge, 1982.
2
[3] J. L. Kelley, "General Topology", Van Nostrand, 1955.
3
[4] S. Mac Lane, "Categories for the Working Mathematician", Springer-Verlag, New York, 1971.
4
[5] J. Picado and A. Pultr, "Locales treated mostly in a covariant way", Textos de Matem'{a}tica, Vol. 41, University of Coimbra, 2008.
5
[6] J. Picado and A. Pultr, "Frames and Locales: topology without points", Frontiers in Mathematics, Vol. 28, Springer, Basel, 2012.
6
[7] T. Plewe, Quotient maps of locales, Appl. Categ. Structures 8 (2000), 17-44.
7
ORIGINAL_ARTICLE
The $\lambda$-super socle of the ring of continuous functions
The concept of $\lambda$-super socle of $C(X)$, denoted by $S_\lambda(X)$ (i.e., the set of elements of $C(X)$ such that the cardinality of their cozerosets are less than $\lambda$, where $\lambda$ is a regular cardinal number with $\lambda\leq |X|$) is introduced and studied. Using this concept we extend some of the basic results concerning $SC_F(X)$, the super socle of $C(X)$ to $S_\lambda(X)$, where $\lambda \geq\aleph_0$. In particular, we determine spaces $X$ for which $SC_F(X)$ and $S_\lambda(X)$ coincide. The one-point $\lambda$-compactification of a discrete space is algebraically characterized via the concept of $\lambda$-super socle. In fact we show that $X$ is the one-point $\lambda$-compactification of a discrete space $Y$ if and only if $S_\lambda(X)$ is a regular ideal and $S_\lambda(X)=O_x$, for some $x\in X$.
http://cgasa.sbu.ac.ir/article_33814_ae287573db032d67df112083dcb83c8f.pdf
2017-01-01T11:23:20
2019-10-21T11:23:20
37
50
$lambda$-super socle
$lambda$-isolated point
one point $lambda$-compactification
$p_lambda$-space
Simin
Mehran
s-mehran@phdstu.scu.ac.ir
true
1
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
AUTHOR
Mehrdad
Namdari
namdari@ipm.ir
true
2
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
AUTHOR
[1] Azarpanah, F., Essential ideals in C(X), Period. Math. Hungar. 31 (1995), 105-112.
1
[2] Azarpanah, F., Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125 (1997), 2149-2154.
2
[3] Azarpanah, F., and O.A.S. Karamzadeh, Algebraic characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12 (2002), 155-168.
3
[4] Azarpanah, F., O.A.S. Karamzadeh, Z. Keshtkar, and A.R. Olfati, On maximal ideals of Cc(X) and the uniformity of its localizations, to appear in Rocky Mountain J. Math.
4
[5] Azarpanah, F., O.A.S. Karamzadeh, and S. Rahmati, C(X) vs. C(X) modulo its socle, Coll. Math. 3 (2008), 315-336.
5
[6] Banaschewski, B., Pointfree topology and the spectra of f-rings, Ordered Algebraic Structures (Proc. Curaccao Conference, 1995), Kluwer Academic Publishers, Dordrecht (1997), 123-148.
6
[7] Ciesielski, K., Set Theory for the Working Mathematician", Cambridge University Press, 1997.
7
[8] Dube, T., Contracting the socle in rings of continuous functions, Rend. Semin. Mat. Univ. Padova 123 (2010), 37-53.
8
[9] Engelking, R., General Topology", Heldermann Verlag Berlin, 1989.
9
[10] Estaji, A.A., and O.A.S. Karamzadeh, On C(X) modulo its socle, Comm. Algebra 13 (2003), 1561-1571.
10
[11] Ghasemzadeh, S., O.A.S. Karamzadeh, and M. Namdari, The super socle of the ring of countinuous functions, to appear in Math. Slovaca.
11
[12] Ghadermazi, M., O.A.S. Karamzadeh, and M. Namdari, C(X) versus its functionally countable subalgebra, submitted to Colloq. Math.
12
[13] Ghadermazi, M., O.A.S. Karamzadeh, and M. Namdari, On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova 129 (2013), 47-70.
13
[14] Gillman, L., and M. Jerison, Rings of continuous functions", Springer-Verlag, 1976.
14
[15] Goodearl, K.R., and R.B. Warfield JR., An Introduction to Noncommutative Noetherian Rings", Cambridge University Press, 1989.
15
[16] Goodearl, K. R., Von Neumann Regular Rings", Pitman Publishing Limited, 1979.
16
[17] Karamzadeh, O.A.S., M. Namdari, and M.A. Siavoshi, A note on lambda-compact spaces, Math. Slovaca 63(6) (2013), 1371-1380.
17
[18] Karamzadeh, O.A.S., M. Namdari, and S. Soltanpour, On the locally functionally countable subalgebra of C(X), Appl. Gen. Topol. 16(2) (2015), 183-207.
18
[19] Karamzadeh, O.A.S., and M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Amer. Math. Soc. 93 (1985), 179-184.
19
[20] Namdari, M., and M.A. Siavoshi, A generalization of compact spaces, JP J. Geom. Topol. 11 (2011), 259-270.
20
[21] Siavoshi, M.A., On lambda-compact spaces, Ph.D. Thesis, Shahid Chamran University of Ahvaz, (2012)
21
ORIGINAL_ARTICLE
C-connected frame congruences
We discuss the congruences $\theta$ that are connected as elements of the (totally disconnected) congruence frame $\CF L$, and show that they are in a one-to-one correspondence with the completely prime elements of $L$, giving an explicit formula. Then we investigate those frames $L$ with enough connected congruences to cover the whole of $\CF L$. They are, among others, shown to be $T_D$-spatial; characteristics for some special cases (Boolean, linear, scattered and Noetherian) are presented.
http://cgasa.sbu.ac.ir/article_34405_5a2102b74343052718d395077541ad72.pdf
2017-01-01T11:23:20
2019-10-21T11:23:20
51
66
frame
frame congruence
congruence and sublocale lattice
connectedness
$T_D$-spatiality
Dharmanand
Baboolal
baboolald@ukzn.ac.za
true
1
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
AUTHOR
Paranjothi
Pillay
pillaypi@ukzn.ac.za
true
2
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.
AUTHOR
Ales
Pultr
pultr@kam.ms.mff.cuni.cz
true
3
Department of Applied Mathematics and CE-ITI, MFF, Charles University, Malostransk\'e n\'am. 24, 11800 Praha 1, Czech Republic.
Department of Applied Mathematics and CE-ITI, MFF, Charles University, Malostransk\'e n\'am. 24, 11800 Praha 1, Czech Republic.
Department of Applied Mathematics and CE-ITI, MFF, Charles University, Malostransk\'e n\'am. 24, 11800 Praha 1, Czech Republic.
AUTHOR
[1] Aull, C.E. and W.J. Thron, Separation axioms between T0 and T1, Indag. Math. 24 (1962), 26-37.
1
[2] Ball, R.N., J. Picado, and A. Pultr, On an aspect of scatteredness in the point-free setting, Port. Math. 73(2) (2016), 139-152.
2
[3] Banaschewski, B., J.L. Frith, and C.R.A. Gilmour, On the congruence lattice of a frame, Pacific J. Math. 130(2) (1987), 209-213.
3
[4] Banaschewski, B. and A. Pultr, Pointfree aspects of the TD axiom of classical topology, Quaest. Math. 33(3) (2010), 369-385.
4
[5] Birkhof, G., Lattice Theory", Amer. Math. Soc. Colloq. Publ. Vol. 25, Third edition, American Mathematical Society, 1967.
5
[6] Chen, X., On the local connectedness of frames, J. Pure Appl. Algebra 79 (1992), 35-43.
6
[7] Dube, T., Submaximality in locales, Topology Proc. 29 (2005), 431-444.
7
[8] Gratzer, G., General Lattice Theory", Academic Press, 1978.
8
[9] Isbell, J.R., Atomless parts of spaces, Math. Scand. 31 (1972), 5-32.
9
[10] Johnstone, P.T., Stone Spaces", Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Cambridge, 1982.
10
[11] Picado, J. and A. Pultr, Frames and Locales: Topology without Points", Frontiers in Mathematics 28, Springer, Basel, 2012.
11
[12] Picado, J. and A. Pultr, Still more about subfitness, to appear in Appl. Categ. Structures.
12
[13] Plewe, T., Sublocale lattices, J. Pure and Appl. Algebra 168 (2002), 309-326.
13
ORIGINAL_ARTICLE
Slimming and regularization of cozero maps
Cozero maps are generalized forms of cozero elements. Two particular cases of cozero maps, slim and regular cozero maps, are significant. In this paper we present methods to construct slim and regular cozero maps from a given cozero map. The construction of the slim and the regular cozero map from a cozero map are called slimming and regularization of the cozero map, respectively. Also, we prove that the slimming and regularization create reflector functors, and so we may say that they are the best method of constructing slim and regular cozero maps, in the sense of category theory. Finally, we give slim regularization for a cozero map $c:M\rightarrow L$ in the general case where $A$ is not a ${\Bbb Q}$-algebra. We use the ring and module of fractions, in this construction process.
http://cgasa.sbu.ac.ir/article_34407_a5c130e088026ede497dc3f85308de65.pdf
2017-01-01T11:23:20
2019-10-21T11:23:20
67
84
frame
cozero map
slim
slimming
algebraic
regular
Regularization
Mohamad Mehdi
Ebrahimi
m-ebrahimi@sbu.ac.ir
true
1
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.
AUTHOR
Abolghasem Karimi
Feizabadi
akarimi@gorganiau.ac.ir
true
2
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
AUTHOR
[1] Adamek, J., H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories", John Wiley and Sons, Inc., 1990.
1
[2] Ball, R.N. and A. Pultr, Quotients and colimits of kappa-quantales, Topology Appl. 158 (2011), 2294-2306.
2
[3] Banaschewski, B., Propositional logic, frames, and fuzzy algebra, Quaest. Math. 22 (1999), 481-508.
3
[4] Banaschewski, B., Pointfree topology and the spectra of f-rings, Ordered algebraic structures (Curaccao, 1995), Kluwer Acad. Publ., Dordrecht, (1997), 123-148.
4
[5] Banaschewski, B., The real numbers in pointfree topology, Texts in Mathematics (Series B), 12, University of Coimbra, 1997.
5
[6] Banaschewski, B. and C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37(3) (1996), 577-587.
6
[7] Banaschewski, B. and C. Gilmour, Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9(4) (2001), 395-417.
7
[8] Banaschewski, B. and C. Gilmour, Cozero bases of frames, J. Pure Appl. Algebra 157(1) (2001), 1-22.
8
[9] Dube, T., Concerning the frame of minimal prime ideals of pointfree function rings, Categ. Gen. Algebr. Struct. Appl. 1(1) (2013), 11-26.
9
[10] Ebrahimi, M.M. and A. Karimi Feizabadi, Spectra of `-Modules, J. Pure Appl. Algebra 208 (2007), 53-60.
10
[11] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree version of Kakutani duality, Order, 22 (2005), 241-256.
11
[12] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291-299.
12
[13] Ebrahimi, M.M., A. Karimi, and M. Mahmoudi, Pointfree spectra of Riesz spaces, Appl. Categ. Structures 12 (2004), 397-409.
13
[14] Gillman, L. and M. Jerison, Rings of Continuous Function", Graduate Texts in Mathematics 43, Springer-Verlag, 1979.
14
[15] Gutierrez Garccia, J., J. Picado, and A. Pultr, Notes on point-free real functions and sublocales, Categorical Methods in Algebra and Topology, Textos de Matematica, 46, University of Coimbra, (2014), 167-200.
15
[16] Karimi Feizabadi, A., Representation of slim algebraic regular cozero maps, Quaest. Math. 29 (2006), 383-394.
16
[17] Johnstone, P.T., Stone Spaces", Cambridge University Press, 1982.
17
[18] Matutu, P., The cozero part of a biframe, Kyungpook Math. J., 42(2) (2002), 285-295.
18
ORIGINAL_ARTICLE
Span and cospan representations of weak double categories
We prove that many important weak double categories can be `represented' by spans, using the basic higher limit of the theory: the tabulator. Dually, representations by cospans via cotabulators are also frequent.
http://cgasa.sbu.ac.ir/article_39606_b35c476a103000cd063c479026904e91.pdf
2017-01-01T11:23:20
2019-10-21T11:23:20
85
105
Double category
tabulator
span
Marco
Grandis
grandis@dima.unige.it
true
1
Dipartimento di Matematica, Universit\`a di Genova, Via Dodecaneso 35,
16146-Genova, Italy
Dipartimento di Matematica, Universit\`a di Genova, Via Dodecaneso 35,
16146-Genova, Italy
Dipartimento di Matematica, Universit\`a di Genova, Via Dodecaneso 35,
16146-Genova, Italy
AUTHOR
Robert
Par\'e
r.pare@dal.ca
true
2
Department of Mathematics and Statistics, Dalhousie University,
Halifax NS, Canada B3H 4R2
Department of Mathematics and Statistics, Dalhousie University,
Halifax NS, Canada B3H 4R2
Department of Mathematics and Statistics, Dalhousie University,
Halifax NS, Canada B3H 4R2
AUTHOR
[1] Benabou, J., Introduction to Bicategories", in: Reports of the Midwest Category Seminar, Lecture Notes in Math. 47, Springer, Berlin 1967, 1-77.
1
[2] Ehresmann, C., Categories structurees, Ann. Sci. Ec. Norm. Super. 80 (1963), 349-425.
2
[3] Ehresmann, C., Categories et structures", Dunod, Paris, 1965.
3
[4] Grandis, M. and R. Pare, Limits in double categories, Cah. Topol. Geom. Differ. Categ. 40 (1999), 162-220.
4
[5] Grandis, M. and R. Pare, Adjoint for double categories, Cah. Topol. Geom. Differ. Categ. 45 (2004), 193-240.
5
[6] Grandis, M. and R. Pare, Kan extensions in double categories (On weak double categories, III), Theory Appl. Categ. 20(8) (2008), 152-185.
6
[7] Grandis, M. and R. Pare, Lax Kan extensions for double categories (On weak double categories, Part IV), Cah. Topol. Geom. Differ. Categ. 48 (2007), 163-199.
7
[8] Niefeld S., Span, cospan, and other double categories, Theory Appl. Categ. 26(26) (2012), 729-742.
8
[9] Street, R., Limits indexed by category-valued 2-functors, J. Pure Appl. Alg. 8 (1976), 149-181.
9
ORIGINAL_ARTICLE
On MV-algebras of non-linear functions
In this paper, the main results are:a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I;a study of Hopfian MV-algebras; anda category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism).
http://cgasa.sbu.ac.ir/article_40443_c18b8b2c7dd7e06904930c09a08f4a60.pdf
2017-01-01T11:23:20
2019-10-21T11:23:20
107
120
MV-algebra
McNaughton function
Hopfian algebra
Antonio
Di Nola
adinola@unisa.it
true
1
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
AUTHOR
Giacomo
Lenzi
gilenzi@unisa.it
true
2
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
LEAD_AUTHOR
Gaetano
Vitale
gvitale@unisa.it
true
3
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I-84084 Fisciano (SA), Italy.
AUTHOR
[1] Belluce, L.P., Di Nola, A., and Lenzi, G., Hyperfinite MV-algebras, J. Pure Appl. Algebra 217 (2013), 1208-1223.
1
[2] Birkhof G. and Pierce, R., Lattice-ordered rings, An. Acad. Brasil. Ci^enc. 28 (1956), 41-69.
2
[3] Cabrer, L. and Mundici, D., Projective MV-algebras and rational polyhedra, Algebra Universalis 62 (2009), 63-74.
3
[4] Cignoli, R., D'Ottaviano, I., and Mundici, D., Algebraic Foundations of Many-Valued Reasoning", Trends in Logic-Studia Logica Library 7, Springer, 2000.
4
[5] Evans, T., Finitely presented loops, lattices, etc. are hopfian, J. London Math. Soc. 44 (1969), 551-552.
5
[6] Hopf, H., Fundamentalgruppe und zweite Bettische Gruppe (German), Comment. Math. Helv. 14 (1942), 257-309.
6
[7] Loats, J. and Rubin, M., Boolean algebras without nontrivial onto endomorphisms exist in every uncountable cardinality, Proc. Amer. Math. Soc. 72(2) (1978), 346-351.
7
[8] Lyapunov, A.M., The general problem of the stability of motion, Translated by A.T. Fuller from Edouard Davaux's French translation (1907) of the 1892 Russian original, Int. J. Control 55(3) (1992), 531-773.
8
[9] Marra, V. and Spada, L., The dual adjunction between MV-algebras and Tychonof spaces, Studia Logica 100(1-2) (2012), 253-278.
9
[10] Mundici, D., Personal communication.
10
[11] Mundici, D., Advanced Lukasiewicz Calculus and MV-algebras", Trends in Logic-Studia Logica Library 35, Springer, 2011.
11
[12] Mundici, D., Hopfian `-groups, MV-algebras and AF C-algebras, ArXiv:1509.03230.
12
[13] Peano, G., Sur une courbe, qui remplit toute une aire plane, Math. Ann. 36(1) (1890), 157-160.
13
[14] Schoenberg, I.J., Contributions to the problem of approximation of equidistant data by analytic functions, Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae, Quart. Appl. Math. 4 (1946), 45-99.
14
[15] Schoenberg, I.J., Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae, Quart. Appl. Math. 4 (1946), 112-141.
15
[16] Steinberg, B., Private communication.
16
[17] Verhulst, P.F., Recherches mathematiques sur la loi d'accroissement de la population, NouveauxMemoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles 18 (1845), 14-54.
17
ORIGINAL_ARTICLE
Choice principles and lift lemmas
We show that in ${\bf ZF}$ set theory without choice, the Ultrafilter Principle (${\bf UP}$) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasicontinuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from ${\bf UP}$ but also from ${\bf DC}$, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets,which is not provable in ${\bf ZF}$ set theory.
http://cgasa.sbu.ac.ir/article_40448_f354e76a770fa82f66fa30955e1aba56.pdf
2017-01-01T11:23:20
2019-10-21T11:23:20
121
146
Choice
(super)compact
foot
free semilattice
locale
noetherian
prime
sober
well-filtered
Marcel
Ern\'e
erne@math.uni-hannover.de
true
1
Faculty for Mathematics and Physics, IAZD, Leibniz Universit\"at, Welfengarten 1, D 30167 Hannover, Germany.
Faculty for Mathematics and Physics, IAZD, Leibniz Universit\"at, Welfengarten 1, D 30167 Hannover, Germany.
Faculty for Mathematics and Physics, IAZD, Leibniz Universit\"at, Welfengarten 1, D 30167 Hannover, Germany.
AUTHOR
[1] Alexandrof, P., Diskrete Raume, Mat. Sb. (N.S.) 2 (1937), 501-518.
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[2] Banaschewski, B., Coherent frames, in: B. Banaschewski and R.-E. Hofmann (eds.), Continuous Lattices, Lecture Notes in Math. 871, Springer, Berlin, 1981, 1-11.
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[3] Banaschewski, B., The power of the ultrafilter theorem, J. London Math. Soc. 27(2) (1983), 193-202.
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[4] Banaschewski, B., Prime elements from prime ideals, Order 2 (1985), 211-213.
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[5] Banaschewski, B. and Erne, M., On Krull's separation lemma, Order 10 (1993), 253-260.
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[6] Banaschewski, B. and Harting, R., Lattice aspects of radical ideals and choice principles, Proc. London Math. Soc. 50 (1985), 385-404.
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[7] Birkhof, G., Lattice Theory", Amer. Math. Soc. Coll. Publ. 25, Providence, R.I., 1st ed. 1948, 3d ed. 1973.
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[8] Brunner, N., Sequential compactness and the axiom of choice, Notre Dame J. Form. Log. 24 (1983), 89-92.
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[9] Erne, M., Einfuhrung in die Ordnungstheorie", B.I.-Wissenschaftsverlag, Bibliographisches Institut, Mannheim, 1982.
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[10] Erne, M., Order, Topology and Closure", University of Hannover, 1982.
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[11] Erne, M., On the existence of decompositions in lattices, Algebra Universalis 16 (1983), 338-343.
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[12] Erne, M., A strong version of the Prime Element Theorem, Preprint, University of Hannover, 1986.
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[13] Erne, M., Ordnungs- und Verbandstheorie", Fernuniversitat Hagen, 1987.
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[14] Erne, M., The ABC of order and topology, in: H. Herrlich and H.-E. Porst (eds.), Category Theory at Work", Heldermann, Berlin, 1991, 57-83.
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[15] Erne, M., Prime ideal theorems and systems of finite character, Comment. Math. Univ. Carolinae 38 (1997), 513-536.
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[16] Erne, M., Prime ideal theory for general algebras, Appl. Categ. Structures 8 (2000), 115-144.
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[17] Erne, M., Minimal bases, ideal extensions, and basic dualities, Topology Proc. 29 (2005), 445-489.
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[18] Erne, M., Choiceless, pointless, but not useless: dualities for preframes, Appl. Categ. Structures 15 (2007), 541-572.
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[19] Erne, M., Infinite distributive laws versus local connectedness and compactness properties, Topology Appl. 156 (2009) 2054-2069.
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[20] Erne, M., The strength of prime ideal separation, sobriety, and compactness theorems, Preprint, Leibniz University Hannover, 2016. See also: Erne, M., Sober spaces, well-filtration and compactness principles, http://www.iazd.uni-hannover.de/ erne/preprints/sober.pdf (2007).
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[21] Felscher, W., Naive Mengen und abstrakte Zahlen" III, B.I. Wissenschaftsverlag, Mannheim, 1979.
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[22] Frasse, R., Theory of Relations", Studies in Logic 118, North-Holland, Amsterdam, 1986.
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[23] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., Continuous Lattices and Domains", Oxford University Press, 2003.
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[24] Gierz, G., Lawson, J.D., Stralka, A.R., Quasicontinuous posets, Houston J. Math. 9 (1983), 191-208.
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[25] Heckmann, R. and Keimel, K., Quasicontinuous domains and the Smyth powerdomain, Electron. Notes Theor. Comput. Sci. 298 (2013), 215-232.
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[26] Herrlich, H., Axiom of Choice", Lecture Notes in Math. 1876, Springer, Berlin Heidelberg, 2006.
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[27] Hoft, H. and Howard, P., Well-ordered subsets of linearly ordered sets, Notre Dame J. Form. Log. 35 (1994), 413-425.
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[28] Howard, P. and Rubin, J.E., Consequences of the Axiom of Choice", AMS Mathematical Surveys and Monographs 59, Providence, 1998.
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[29] Isbell, J.R., Function spaces and adjoints, Math. Scand. 36 (1975), 317-339.
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[30] Jech, T.J., The Axiom of Choice", North-Holland, Amsterdam, 1973.
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[31] Johnstone, P.T., Scott is not always sober, in: B. Banaschewski and R.-E. Hofmann (eds.), Continuous Lattices", Lecture Notes in Math. 871, Springer, Berlin, 1981, 282-283.
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[32] Johnstone, P.T., Stone Spaces", Cambridge University Press, 1982.
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[33] Jung, A., Cartesian Closed Categories of Domains, CWI Tracts 66, Centrum voor Wiskunde en Informatica, Amsterdam (1989), 107 pp.
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[34] Konig, D., Uber eine Schlussweise aus dem Endlichen ins Unendliche: Punktmengen. Kartenfarben. Verwandtschaftsbeziehungen. Schachspiel, Acta Lit. Sci. Reg. Univ. Hung. 3 (1927), 121-130.
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[35] Krom, M., Equivalents of a weak axiom of choice, Notre Dame J. Form. Log. 22 (1981), 283-285.
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[36] Moore, G.H., Zermelo's Axiom of Choice", Springer, Berlin Heidelberg NewYork, 1982.
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[37] Picado, J. and Pultr, A., Frames and Locales", Birkhauser, Basel, 2012.
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[38] Rubin, H., and Scott, D.S., Some topological theorems equivalent to the prime ideal theorem, Bull. Amer. Math. Soc. 60 (1954), 389 (Abstract).
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[39] Rudin, M., Directed sets which converge, in: McAuley, L.F., and Rao, M.M. (eds.), General Topology and Modern Analysis", University of California, Riverside, 1980, Academic Press, 1981, 305-307.
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[40] Scott, D.S., Prime ideal theorems for rings, lattices and Boolean algebras, Bull. Amer. Math. Soc. 60 (1954), 390 (Abstract).
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[41] Tarski, A., Prime ideal theorems for Boolean algebras and the axiom of choice, Bull. Amer. Math. Soc. 60 (1954), 390-391 (Abstract).
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[42] Tarski, A., Algebraic and axiomatic aspects of two theorems on sums of cardinals, Fund. Math. 35 (1948), 79-104.
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[43] Wyler, O., Dedekind complete posets and Scott topologies, in: B. Banaschewski and R.E. Hofmann (eds.), Continuous Lattices", Lecture Notes in Math. 871, pringer, Berlin, 1981, 384-389.
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