2017
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Everyday physics of extended bodies or why functionals need analyzing
2
2
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 nontechnical 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 infinitedimensional 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.
1

9
19


F. William
Lawvere
Professor Emeritus, Department of Mathematics, University at Buffalo, Buffalo, New York 142602900, United States of America.
Professor Emeritus, Department of Mathematics,
Iran
wlawvere@buffalo.edu
Functionals
physics
[[1] Bell, J.L., A Primer of Infinitesimal Analysis", Cambridge University Press, Cambridge,1998.##[2] Kock, A., Synthetic Differential Geometry", Cambridge University Press (2nd Edition),Cambridge, 2006.##[3] Lavendhomme, R., Basic Concepts of Synthetic Differential Geometry", KluwerAcademic Publishers Dordrecht, Kluwer Texts in Mathematical Sciences 13, 1996.##[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), 337392.##[5] Lawvere, F.W., Comments on the development of topos theory, in the book:Development of Mathematics 19502000", J.P. Pier (Ed) Birkhauser Verlag, Basel,(2000), 715734. See also in TAC Reprints 24 (2014), 122 (with author commentary).##[6] Moerdijk, I. and Reyes, G., Models for Smooth Infinitesimal Analysis", SpringerVerlag, New York, 1991.##]
Localic maps constructed from open and closed parts
2
2
Assembling a localic map $fcolon Lto M$ from localic maps $f_icolon S_ito M$, $iin J$, 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 pointfree 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.
1

21
35


Ales
Pultr
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,
Iran
pultr@kam.ms.mff.cuni.cz


Jorge
Picado
CMUC, Department of Mathematics, University of Coimbra, Apartado 3008, 3001501 Coimbra, Portugal.
CMUC, Department of Mathematics, University
Iran
picado@mat.uc.pt
Frame
locale
sublocale
sublocale lattice
open sublocale
closed sublocale
localic map
preimage
Boolean frame
linear frame
[[1] J. R. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972), 532.##[2] P. T. Johnstone, "Stone Spaces", Cambridge Univ. Press, Cambridge, 1982.##[3] J. L. Kelley, "General Topology", Van Nostrand, 1955.##[4] S. Mac Lane, "Categories for the Working Mathematician", SpringerVerlag, New York, 1971.##[5] J. Picado and A. Pultr, "Locales treated mostly in a covariant way", Textos de Matem'{a}tica, Vol. 41, University of Coimbra, 2008.##[6] J. Picado and A. Pultr, "Frames and Locales: topology without points", Frontiers in Mathematics, Vol. 28, Springer, Basel, 2012.##[7] T. Plewe, Quotient maps of locales, Appl. Categ. Structures 8 (2000), 1744.##]
The $lambda$super socle of the ring of continuous functions
2
2
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 $lambdaleq X$) 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 $lambda geqaleph_0$. In particular, we determine spaces $X$ for which $SC_F(X)$ and $S_lambda(X)$ coincide. The onepoint $lambda$compactification of a discrete space is algebraically characterized via the concept of $lambda$super socle. In fact we show that $X$ is the onepoint $lambda$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 $xin X$.
1

37
50


Simin
Mehran
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Department of Mathematics, Shahid Chamran
Iran
smehran@phdstu.scu.ac.ir


Mehrdad
Namdari
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Department of Mathematics, Shahid Chamran
Iran
namdari@ipm.ir
$lambda$super socle
$lambda$isolated point
one point $lambda$compactification
$p_lambda$space
[[1] Azarpanah, F., Essential ideals in C(X), Period. Math. Hungar. 31 (1995), 105112.##[2] Azarpanah, F., Intersection of essential ideals in C(X), Proc. Amer. Math. Soc. 125 (1997), 21492154.##[3] Azarpanah, F., and O.A.S. Karamzadeh, Algebraic characterization of some disconnected spaces, Italian. J. Pure Appl. Math. 12 (2002), 155168.##[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.##[5] Azarpanah, F., O.A.S. Karamzadeh, and S. Rahmati, C(X) vs. C(X) modulo its socle, Coll. Math. 3 (2008), 315336.##[6] Banaschewski, B., Pointfree topology and the spectra of frings, Ordered Algebraic Structures (Proc. Curaccao Conference, 1995), Kluwer Academic Publishers, Dordrecht (1997), 123148.##[7] Ciesielski, K., Set Theory for the Working Mathematician", Cambridge University Press, 1997.##[8] Dube, T., Contracting the socle in rings of continuous functions, Rend. Semin. Mat. Univ. Padova 123 (2010), 3753.##[9] Engelking, R., General Topology", Heldermann Verlag Berlin, 1989.##[10] Estaji, A.A., and O.A.S. Karamzadeh, On C(X) modulo its socle, Comm. Algebra 13 (2003), 15611571.##[11] Ghasemzadeh, S., O.A.S. Karamzadeh, and M. Namdari, The super socle of the ring of countinuous functions, to appear in Math. Slovaca.##[12] Ghadermazi, M., O.A.S. Karamzadeh, and M. Namdari, C(X) versus its functionally countable subalgebra, submitted to Colloq. Math.##[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), 4770.##[14] Gillman, L., and M. Jerison, Rings of continuous functions", SpringerVerlag, 1976.##[15] Goodearl, K.R., and R.B. Warfield JR., An Introduction to Noncommutative Noetherian Rings", Cambridge University Press, 1989.##[16] Goodearl, K. R., Von Neumann Regular Rings", Pitman Publishing Limited, 1979.##[17] Karamzadeh, O.A.S., M. Namdari, and M.A. Siavoshi, A note on lambdacompact spaces, Math. Slovaca 63(6) (2013), 13711380.##[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), 183207.##[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), 179184.##[20] Namdari, M., and M.A. Siavoshi, A generalization of compact spaces, JP J. Geom. Topol. 11 (2011), 259270.##[21] Siavoshi, M.A., On lambdacompact spaces, Ph.D. Thesis, Shahid Chamran University of Ahvaz, (2012)##]
Cconnected frame congruences
2
2
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 onetoone 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.
1

51
66


Dharmanand
Baboolal
School of Mathematics, Statistics and Computer Science, University of KwaZuluNatal, Durban 4000, South Africa.
School of Mathematics, Statistics and Computer
Iran
baboolald@ukzn.ac.za


Paranjothi
Pillay
School of Mathematics, Statistics and Computer Science, University of KwaZuluNatal, Durban 4000, South Africa.
School of Mathematics, Statistics and Computer
Iran
pillaypi@ukzn.ac.za


Ales
Pultr
Department of Applied Mathematics and CEITI, MFF, Charles University, Malostransk'e n'am. 24, 11800 Praha 1, Czech Republic.
Department of Applied Mathematics and CEITI,
Iran
pultr@kam.ms.mff.cuni.cz
Frame
frame congruence
congruence and sublocale lattice
connectedness
$T_D$spatiality
[[1] Aull, C.E. and W.J. Thron, Separation axioms between T0 and T1, Indag. Math. 24 (1962), 2637.##[2] Ball, R.N., J. Picado, and A. Pultr, On an aspect of scatteredness in the pointfree setting, Port. Math. 73(2) (2016), 139152.##[3] Banaschewski, B., J.L. Frith, and C.R.A. Gilmour, On the congruence lattice of a frame, Pacific J. Math. 130(2) (1987), 209213.##[4] Banaschewski, B. and A. Pultr, Pointfree aspects of the TD axiom of classical topology, Quaest. Math. 33(3) (2010), 369385.##[5] Birkhof, G., Lattice Theory", Amer. Math. Soc. Colloq. Publ. Vol. 25, Third edition, American Mathematical Society, 1967.##[6] Chen, X., On the local connectedness of frames, J. Pure Appl. Algebra 79 (1992), 3543.##[7] Dube, T., Submaximality in locales, Topology Proc. 29 (2005), 431444.##[8] Gratzer, G., General Lattice Theory", Academic Press, 1978.##[9] Isbell, J.R., Atomless parts of spaces, Math. Scand. 31 (1972), 532.##[10] Johnstone, P.T., Stone Spaces", Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Cambridge, 1982.##[11] Picado, J. and A. Pultr, Frames and Locales: Topology without Points", Frontiers in Mathematics 28, Springer, Basel, 2012.##[12] Picado, J. and A. Pultr, Still more about subfitness, to appear in Appl. Categ. Structures.##[13] Plewe, T., Sublocale lattices, J. Pure and Appl. Algebra 168 (2002), 309326.##]
Slimming and regularization of cozero maps
2
2
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 $c:Mrightarrow L$ in the general case where $A$ is not a ${Bbb Q}$algebra. We use the ring and module of fractions, in this construction process.
1

67
84


Mohamad Mehdi
Ebrahimi
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.
Department of Mathematics, Shahid Beheshti
Iran
mebrahimi@sbu.ac.ir


Abolghasem Karimi
Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
Department of Mathematics, Gorgan Branch,
Iran
akarimi@gorganiau.ac.ir
Frame
cozero map
slim
slimming
algebraic
regular
Regularization
[[1] Adamek, J., H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories", John Wiley and Sons, Inc., 1990.##[2] Ball, R.N. and A. Pultr, Quotients and colimits of kappaquantales, Topology Appl. 158 (2011), 22942306.##[3] Banaschewski, B., Propositional logic, frames, and fuzzy algebra, Quaest. Math. 22 (1999), 481508.##[4] Banaschewski, B., Pointfree topology and the spectra of frings, Ordered algebraic structures (Curaccao, 1995), Kluwer Acad. Publ., Dordrecht, (1997), 123148.##[5] Banaschewski, B., The real numbers in pointfree topology, Texts in Mathematics (Series B), 12, University of Coimbra, 1997.##[6] Banaschewski, B. and C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37(3) (1996), 577587.##[7] Banaschewski, B. and C. Gilmour, Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9(4) (2001), 395417.##[8] Banaschewski, B. and C. Gilmour, Cozero bases of frames, J. Pure Appl. Algebra 157(1) (2001), 122.##[9] Dube, T., Concerning the frame of minimal prime ideals of pointfree function rings, Categ. Gen. Algebr. Struct. Appl. 1(1) (2013), 1126.##[10] Ebrahimi, M.M. and A. Karimi Feizabadi, Spectra of `Modules, J. Pure Appl. Algebra 208 (2007), 5360.##[11] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree version of Kakutani duality, Order, 22 (2005), 241256.##[12] Ebrahimi, M.M. and A. Karimi Feizabadi, Pointfree prime representation of real Riesz maps, Algebra Universalis 54 (2005), 291299.##[13] Ebrahimi, M.M., A. Karimi, and M. Mahmoudi, Pointfree spectra of Riesz spaces, Appl. Categ. Structures 12 (2004), 397409.##[14] Gillman, L. and M. Jerison, Rings of Continuous Function", Graduate Texts in Mathematics 43, SpringerVerlag, 1979.##[15] Gutierrez Garccia, J., J. Picado, and A. Pultr, Notes on pointfree real functions and sublocales, Categorical Methods in Algebra and Topology, Textos de Matematica, 46, University of Coimbra, (2014), 167200.##[16] Karimi Feizabadi, A., Representation of slim algebraic regular cozero maps, Quaest. Math. 29 (2006), 383394.##[17] Johnstone, P.T., Stone Spaces", Cambridge University Press, 1982.##[18] Matutu, P., The cozero part of a biframe, Kyungpook Math. J., 42(2) (2002), 285295.##]
Span and cospan representations of weak double categories
2
2
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.
1

85
105


Marco
Grandis
Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35,
16146Genova, Italy
Dipartimento di Matematica, Universit`a
Iran
grandis@dima.unige.it


Robert
Par'e
Department of Mathematics and Statistics, Dalhousie University,
Halifax NS, Canada B3H 4R2
Department of Mathematics and Statistics,
Iran
r.pare@dal.ca
Double category
tabulator
span
[[1] Benabou, J., Introduction to Bicategories", in: Reports of the Midwest Category Seminar, Lecture Notes in Math. 47, Springer, Berlin 1967, 177.##[2] Ehresmann, C., Categories structurees, Ann. Sci. Ec. Norm. Super. 80 (1963), 349425.##[3] Ehresmann, C., Categories et structures", Dunod, Paris, 1965.##[4] Grandis, M. and R. Pare, Limits in double categories, Cah. Topol. Geom. Differ. Categ. 40 (1999), 162220.##[5] Grandis, M. and R. Pare, Adjoint for double categories, Cah. Topol. Geom. Differ. Categ. 45 (2004), 193240.##[6] Grandis, M. and R. Pare, Kan extensions in double categories (On weak double categories, III), Theory Appl. Categ. 20(8) (2008), 152185.##[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), 163199.##[8] Niefeld S., Span, cospan, and other double categories, Theory Appl. Categ. 26(26) (2012), 729742.##[9] Street, R., Limits indexed by categoryvalued 2functors, J. Pure Appl. Alg. 8 (1976), 149181.##]
On MValgebras of nonlinear functions
2
2
In this paper, the main results are:a study of the finitely generated MValgebras of continuous functions from the nth power of the unit real interval I to I;a study of Hopfian MValgebras; anda categorytheoretic study of the map sending an MValgebra as above to the range of its generators (up to a suitable form of homeomorphism).
1

107
120


Antonio
Di Nola
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I84084 Fisciano (SA), Italy.
Department of Mathematics, University of
Iran
adinola@unisa.it


Giacomo
Lenzi
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I84084 Fisciano (SA), Italy.
Department of Mathematics, University of
Iran
gilenzi@unisa.it


Gaetano
Vitale
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, I84084 Fisciano (SA), Italy.
Department of Mathematics, University of
Iran
gvitale@unisa.it
MValgebra
McNaughton function
Hopfian algebra
[[1] Belluce, L.P., Di Nola, A., and Lenzi, G., Hyperfinite MValgebras, J. Pure Appl. Algebra 217 (2013), 12081223.##[2] Birkhof G. and Pierce, R., Latticeordered rings, An. Acad. Brasil. Ci^enc. 28 (1956), 4169.##[3] Cabrer, L. and Mundici, D., Projective MValgebras and rational polyhedra, Algebra Universalis 62 (2009), 6374.##[4] Cignoli, R., D'Ottaviano, I., and Mundici, D., Algebraic Foundations of ManyValued Reasoning", Trends in LogicStudia Logica Library 7, Springer, 2000.##[5] Evans, T., Finitely presented loops, lattices, etc. are hopfian, J. London Math. Soc. 44 (1969), 551552.##[6] Hopf, H., Fundamentalgruppe und zweite Bettische Gruppe (German), Comment. Math. Helv. 14 (1942), 257309.##[7] Loats, J. and Rubin, M., Boolean algebras without nontrivial onto endomorphisms exist in every uncountable cardinality, Proc. Amer. Math. Soc. 72(2) (1978), 346351.##[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), 531773.##[9] Marra, V. and Spada, L., The dual adjunction between MValgebras and Tychonof spaces, Studia Logica 100(12) (2012), 253278.##[10] Mundici, D., Personal communication.##[11] Mundici, D., Advanced Lukasiewicz Calculus and MValgebras", Trends in LogicStudia Logica Library 35, Springer, 2011.##[12] Mundici, D., Hopfian `groups, MValgebras and AF Calgebras, ArXiv:1509.03230.##[13] Peano, G., Sur une courbe, qui remplit toute une aire plane, Math. Ann. 36(1) (1890), 157160.##[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), 4599.##[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), 112141.##[16] Steinberg, B., Private communication.##[17] Verhulst, P.F., Recherches mathematiques sur la loi d'accroissement de la population, NouveauxMemoires de l'Academie Royale des Sciences et BellesLettres de Bruxelles 18 (1845), 1454.##]
Choice principles and lift lemmas
2
2
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.
1

121
146


Marcel
Ern'e
Faculty for Mathematics and Physics, IAZD, Leibniz Universit"at, Welfengarten 1, D 30167 Hannover, Germany.
Faculty for Mathematics and Physics, IAZD,
Iran
erne@math.unihannover.de
choice
(super)compact
foot
free semilattice
locale
noetherian
prime
sober
wellfiltered
[[1] Alexandrof, P., Diskrete Raume, Mat. Sb. (N.S.) 2 (1937), 501518.##[2] Banaschewski, B., Coherent frames, in: B. Banaschewski and R.E. Hofmann (eds.), Continuous Lattices, Lecture Notes in Math. 871, Springer, Berlin, 1981, 111.##[3] Banaschewski, B., The power of the ultrafilter theorem, J. London Math. Soc. 27(2) (1983), 193202.##[4] Banaschewski, B., Prime elements from prime ideals, Order 2 (1985), 211213.##[5] Banaschewski, B. and Erne, M., On Krull's separation lemma, Order 10 (1993), 253260.##[6] Banaschewski, B. and Harting, R., Lattice aspects of radical ideals and choice principles, Proc. London Math. Soc. 50 (1985), 385404.##[7] Birkhof, G., Lattice Theory", Amer. Math. Soc. Coll. Publ. 25, Providence, R.I., 1st ed. 1948, 3d ed. 1973.##[8] Brunner, N., Sequential compactness and the axiom of choice, Notre Dame J. Form. Log. 24 (1983), 8992.##[9] Erne, M., Einfuhrung in die Ordnungstheorie", B.I.Wissenschaftsverlag, Bibliographisches Institut, Mannheim, 1982.##[10] Erne, M., Order, Topology and Closure", University of Hannover, 1982.##[11] Erne, M., On the existence of decompositions in lattices, Algebra Universalis 16 (1983), 338343.##[12] Erne, M., A strong version of the Prime Element Theorem, Preprint, University of Hannover, 1986.##[13] Erne, M., Ordnungs und Verbandstheorie", Fernuniversitat Hagen, 1987.##[14] Erne, M., The ABC of order and topology, in: H. Herrlich and H.E. Porst (eds.), Category Theory at Work", Heldermann, Berlin, 1991, 5783.##[15] Erne, M., Prime ideal theorems and systems of finite character, Comment. Math. Univ. Carolinae 38 (1997), 513536.##[16] Erne, M., Prime ideal theory for general algebras, Appl. Categ. Structures 8 (2000), 115144.##[17] Erne, M., Minimal bases, ideal extensions, and basic dualities, Topology Proc. 29 (2005), 445489.##[18] Erne, M., Choiceless, pointless, but not useless: dualities for preframes, Appl. Categ. Structures 15 (2007), 541572.##[19] Erne, M., Infinite distributive laws versus local connectedness and compactness properties, Topology Appl. 156 (2009) 20542069.##[20] Erne, M., The strength of prime ideal separation, sobriety, and compactness theorems, Preprint, Leibniz University Hannover, 2016. See also: Erne, M., Sober spaces, wellfiltration and compactness principles, http://www.iazd.unihannover.de/ erne/preprints/sober.pdf (2007).##[21] Felscher, W., Naive Mengen und abstrakte Zahlen" III, B.I. Wissenschaftsverlag, Mannheim, 1979.##[22] Frasse, R., Theory of Relations", Studies in Logic 118, NorthHolland, Amsterdam, 1986.##[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.##[24] Gierz, G., Lawson, J.D., Stralka, A.R., Quasicontinuous posets, Houston J. Math. 9 (1983), 191208.##[25] Heckmann, R. and Keimel, K., Quasicontinuous domains and the Smyth powerdomain, Electron. Notes Theor. Comput. Sci. 298 (2013), 215232.##[26] Herrlich, H., Axiom of Choice", Lecture Notes in Math. 1876, Springer, Berlin Heidelberg, 2006.##[27] Hoft, H. and Howard, P., Wellordered subsets of linearly ordered sets, Notre Dame J. Form. Log. 35 (1994), 413425.##[28] Howard, P. and Rubin, J.E., Consequences of the Axiom of Choice", AMS Mathematical Surveys and Monographs 59, Providence, 1998.##[29] Isbell, J.R., Function spaces and adjoints, Math. Scand. 36 (1975), 317339.##[30] Jech, T.J., The Axiom of Choice", NorthHolland, Amsterdam, 1973.##[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, 282283.##[32] Johnstone, P.T., Stone Spaces", Cambridge University Press, 1982.##[33] Jung, A., Cartesian Closed Categories of Domains, CWI Tracts 66, Centrum voor Wiskunde en Informatica, Amsterdam (1989), 107 pp.##[34] Konig, D., Uber eine Schlussweise aus dem Endlichen ins Unendliche: Punktmengen. Kartenfarben. Verwandtschaftsbeziehungen. Schachspiel, Acta Lit. Sci. Reg. Univ. Hung. 3 (1927), 121130.##[35] Krom, M., Equivalents of a weak axiom of choice, Notre Dame J. Form. Log. 22 (1981), 283285.##[36] Moore, G.H., Zermelo's Axiom of Choice", Springer, Berlin Heidelberg NewYork, 1982.##[37] Picado, J. and Pultr, A., Frames and Locales", Birkhauser, Basel, 2012.##[38] Rubin, H., and Scott, D.S., Some topological theorems equivalent to the prime ideal theorem, Bull. Amer. Math. Soc. 60 (1954), 389 (Abstract).##[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, 305307.##[40] Scott, D.S., Prime ideal theorems for rings, lattices and Boolean algebras, Bull. Amer. Math. Soc. 60 (1954), 390 (Abstract).##[41] Tarski, A., Prime ideal theorems for Boolean algebras and the axiom of choice, Bull. Amer. Math. Soc. 60 (1954), 390391 (Abstract).##[42] Tarski, A., Algebraic and axiomatic aspects of two theorems on sums of cardinals, Fund. Math. 35 (1948), 79104.##[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, 384389.##]