2014
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Uniformities and covering properties for partial frames (I)
2
2
Partial frames provide a rich context in which to do pointfree structured and unstructured topology. A small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. These axioms are sufficiently general to include as examples bounded distributive lattices, $sigma$frames, $kappa$frames and frames. Reflective subcategories of uniform and nearness spaces and lately coreflective subcategories of uniform and nearness frames have been a topic of considerable interest. In cite{jfas9} an easily implementable criterion for establishing certain coreflections in nearness frames was presented. Although the primary application in that paper was in the setting of nearness frames, it was observed there that similar techniques apply in many categories; we establish here, in this more general setting of structured partial frames, a technique that unifies these. We make use of the notion of a partial frame, which is a meetsemilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting our axiomatization of partial frames, which we call $sels$frames, we add structure, in the form of $sels$covers and nearness, and provide the promised method of constructing certain coreflections. We illustrate the method with the examples of uniform, strong and totally bounded nearness $sels$frames. In Part (II) of this paper, we consider regularity, normality and compactness for partial frames.
1

1
21


John
Frith
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701,
South Africa.
Department of Mathematics and Applied Mathematics,
Iran
john.frith@uct.ac.za


Anneliese
Schauerte
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
Department of Mathematics and Applied Mathematics,
Iran
anneliese.schauerte@uct.ac.za
frame
$sels$frame
$Z$frame
partial frame
$sigma$frame
$kappa$frame
meetsemilattice
nearness
Uniformity
strong inclusion
uniform map
coreflection
$P$approximation
strong
totally bounded
regular
normal
compact
[[1] D. Baboolal and R.G. Ori, Samuel compactication and uniform coreection of nearness frames, Proceedings Symposium on Categorical Topology (1994), University of Cape Town, 1999.##[2] B. Banaschewski, Completion in pointfree topology, Lecture Notes in Math. and Applied Math., University of Cape Town, No. 2 (1996).##[3] B. Banaschewski, Uniform completion in pointfree topology, chapter in Topological and Algebraic Structures in Fuzzy Sets, S.E. Rodabaugh and E.P. Klement (Ed.s), Kluwer Academic Publishers, (2003) 1956.##[4] B. Banaschewski and C.R.A. Gilmour, Realcompactness and the cozero part of a frame, Appl. Categ. Structures 9 (2001), 395417.##[5] B. Banaschewski, S.S. Hong, and A. Pultr, On the completion of nearness frames, Quaest. Math. 21 (1998), 1937.##[6] B. Banaschewski and A. Pultr, A general view of approximation, Appl. Categ. Structures 14 (2006), 165190.##[7] B. Banaschewski and A. Pultr, Cauchy points of uniform and nearness frames, Quaest. Math. 19 (1996), 101127.##[8] T. Dube, A note on complete regularity and normality, Quaest. Math. 19 (1996), 467478. [9] J. Frith and A. Schauerte, A method for constructing coreections for nearness frames, Appl. Categ. Structures (to appear).##[10] J. Frith and A. Schauerte, Uniformities and covering properties for partial frames (II), Categ. General Alg. Struct. Appl. 2(1) (2014), 2335.##[11] P.T. Johnstone, Stone Spaces", Cambridge University Press, Cambridge, 1982. ##[12] J.J. Madden, frames, J. Pure Appl. Algebra 70 (1991), 107127.##[13] I. Naidoo, Aspects of nearness in frames, Quaest. Math. 30 (2007), 133145.##[14] J. Paseka, Covers in generalized frames, in: General Algebra and Ordered Sets (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 8499.##[15] J. Picado and A. Pultr, Frames and Locales", Springer, Basel, 2012. ##[16] J. Picado, A. Pultr, and A. Tozzi, Locales, chapter in Categorical Foundations, MC Pedicchio and W Tholen (eds), Encyclopedia of Mathematics and its Applications 97, Cambridge University Press, Cambridge, (2004) 49101.##[17] S. Vickers, Topology via Logic", Cambridge Tracts in Theoretical Computer Science##5, Cambridge University Press, Cambridge, 1989.##[18] J. Walters, Compactications and uniformities on sigma frames, Comment. Math.##Univ. Carolinae 32(1) (1991), 189198.##[19] E.R. Zenk, Categories of partial frames, Algebra Universalis 54 (2005), 213235.##[20] D. Zhao, On projective Zframes, Canad. Math. Bull. 40(1) (1997), 3946.##]
Uniformities and covering properties for partial frames (II)
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2
This paper is a continuation of [Uniformities and covering properties for partial frames (I)], in which we make use of the notion of a partial frame, which is a meetsemilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting there our axiomatization of partial frames, which we call $sels$frames, we added structure, in the form of $sels$covers and nearness. Here, in the unstructured setting, we consider regularity, normality and compactness, expressing all these properties in terms of $sels$covers. We see that an $sels$frame is normal and regular if and only if the collection of all finite $sels$covers forms a basis for an $sels$uniformity on it. Various results about strong inclusions culminate in the proposition that every compact, regular $sels$frame has a unique compatible $sels$uniformity.
1

23
35


John
Frith
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701,
South Africa.
Department of Mathematics and Applied Mathematics,
Iran
john.frith@uct.ac.za


Anneliese
Schauerte
Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.
Department of Mathematics and Applied Mathematics,
Iran
anneliese.schauerte@uct.ac.za
frame
$sels$frame
$Z$frame
partial frame
$sigma$frame
$kappa$frame
meetsemilattice
nearness
Uniformity
strong inclusion
uniform map
coreflection
$P$approximation
strong
totally bounded
regular
normal
compact
[[1] B. Banaschewski, frames, unpublished manuscript, 1980. Available online at http://mathcs.chapman.edu/CECAT/members/Banaschewski publications.##[2] J. Frith and A. Schauerte, Uniformities and covering properties for partial frames (I), Categ. General Alg. Struct. Appl. 2(1) (2014), 121.##[3] C.R.A. Gilmour, Realcompact spaces and regular frames, Math. Proc. Camb. Phil. Soc. 96 (1984), 7379.##[4] J.J. Madden, frames, J. Pure Appl. Algebra 70 (1991), 107127.##[5] J. Madden and J. Vermeer, Lindelof locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473480.##[6] J. Paseka, Covers in generalized frames, in: General Algebra and Ordered Sets (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc, 8499.##[7] J. Picado and A. Pultr, Frames and Locales", Springer, Basel, 2012. ##[8] E.R. Zenk, Categories of partial frames, Algebra Universalis 54 (2005), 213235.##[9] D. Zhao, On projective Zframes, Canad. Math. Bull. 40(1) (1997), 3946.##]
Quasiprojective covers of right $S$acts
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2
In this paper $S$ is a monoid with a left zero and $A_S$ (or $A$) is a unitary right $S$act. It is shown that a monoid $S$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $S$act is quasiprojective. Also it is shown that if every right $S$act has a unique zero element, then the existence of a quasiprojective cover for each right act implies that every right act has a projective cover.
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37
45


Mohammad
Roueentan
Department of Mathematics, College of
Science, Shiraz University, Shiraz 71454, Iran.
Department of Mathematics, College of
Science,
Iran
m.rooeintan@yahoo.com


Majid
Ershad
Department of Mathematics, College of Science, Shiraz University, Shiraz 71454, Iran.
Department of Mathematics, College of Science,
Iran
ershad@shirazu.ac.ir
Projective
quasiprojective
perfect
semiperfect
cover
[[1] J. Ahsan and K. Saifullah, Completely quasiprojective monoids, Semigroup Forum 38 (1989), 123126.##[2] J. Fountain, Perfect semigroups, Proc. Edinburgh Math. Soc. 20(3) (1976), 8793.##[3] J. Isbell, Perfect monoids, Semigroup Forum 2 (1971), 95118.##[4] R. Khosravi, M. Ershad, and M. Sedaghatjoo, Storngly at and condition (P) covers of acts over monoids, Comm. Algebra 38(12) (2010), 45204530.##[5] M. Kilp, U. Knauer, and A. Mikhalev, Monoids, Acts and Categories, With Application to Wreath Products and Graphs"; Berlin, New York, 2000.##[6] U. Knauer and H. Oltmanns, On Rees weakly projective right acts, J. Math. Sci. 139(4) (2006), 67156722.##[7] U. Knauer and H. Oltmanns, Weak projectivities for Sacts, Proceeding of the Conference on General Algebra and Discrete Math. (postsdam), Aachen (1999), 143159.##[8] M. Mahmoudi and J. Renshaw, On covers of cyclic acts over monoids, Semigroup Forum 77 (2008), 325338.##[9] J. Wei, On a question of Kilp and Knauer, Comm. Algebra 32(6) (2004), 22692272.##]
Dually quasiDe Morgan Stone semiHeyting algebras I. Regularity
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2
This paper is the first of a two part series. In this paper, we first prove that the variety of dually quasiDe Morgan Stone semiHeyting algebras of level 1 satisfies the strongly blended $lor$De Morgan law introduced in cite{Sa12}. Then, using this result and the results of cite{Sa12}, we prove our main result which gives an explicit description of simple algebras(=subdirectly irreducibles) in the variety of regular dually quasiDe Morgan Stone semiHeyting algebras of level 1. It is shown that there are 25 nontrivial simple algebras in this variety. In Part II, we prove, using the description of simples obtained in this Part, that the variety $mathbf{RDQDStSH_1}$ of regular dually quasiDe Morgan Stone semiHeyting algebras of level 1 is the join of the variety generated by the twenty 3element $mathbf{RDQDStSH_1}$chains and the variety of dually quasiDe Morgan Boolean semiHeyting algebrasthe latter is known to be generated by the expansions of the three 4element Boolean semiHeyting algebras. As consequences of this theorem, we present (equational) axiomatizations for several subvarieties of $mathbf{RDQDStSH_1}$. The Part II concludes with some open problems for further investigation.
1

47
64


Hanamantagouda P.
Sankappanavar
Department of Mathematics, State University of New York, New Paltz, NY 12561
Department of Mathematics, State University
Iran
sankapph@newpaltz.edu
Regular dually, quasiDe Morgan, semiHeyting algebra of level 1
dually pseudocomplemented semiHeyting algebra
De Morgan semiHeyting algebra
strongly blended dually quasiDe Morgan Stone semiHeyting algebra
discriminator variety
simple
directly indecomposable
subdirectly irreducible
equational base
[[1] M. Abad, J.M. Cornejo and J.P. Diaz Varela, The variety of semiHeyting algebras##satisfying the equation , Reports on Mathematical Logic##46 (2011), 7590.##[2] M. Abad, J.M. Cornejo and J.P. Daz Varela, The variety generated by semiHeyting##chains, Soft Computing 15 (2011), 721728.##[3] M. Abad and L. Monteiro, Free symmetric Boolean algebras, Revista de la U.M.A.##27 (1976), 207215.##[4] R. Balbes and PH. Dwinger, Distributive Lattices", Univ. of Missouri Press,##Columbia, 1974.##[5] S. Burris and H.P. Sankappanavar, A Course in Universal Algebra", Springer##Verlag, New York, 1981. The free, corrected version (2012) is available online as a##PDF file at math.uwaterloo.ca/fisnburris.##[6] G. Gratzer, Lattice Theory", W.H.Freeman and Co., San Francisco, 1971.##[7] A. Horn, Logic with truth values in a linearly ordered Heyting algebras, J. Symbolic.##Logic 34 (1969), 395408.##[8] B. Jhonsson, Algebras whose congruence lattices are distributive, Math. Scand. 21##(1967), 110121.##[9] V.Yu. Meskhi, A discriminator variety of Heyting algebras with involution, Algebra##i Logika 21 (1982), 537552.##[10] A. Monteiro, Sur les algebres de Heyting symetriques, Portugaliae Mathemaica 39##(1980), 1237.##[11] W. McCune, Prover9 and Mace 4, http://www.cs.unm.edu/mccune/prover9/.##[12] H. Rasiowa, An Algebraic Approach to NonClassical Logics", North{Holland##Publ.Comp., Amsterdam, 1974.##[13] H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics", Warsazawa,##[14] H.P. Sankappanavar, Heyting algebras with dual pseudocomplementation, Pacific J.##Math. 117 (1985), 405415.##[15] H.P. Sankappanavar, Pseudocomplemented Okham and De Morgan algebras,##Zeitschr. f. math. Logik und Grundlagen d. Math. 32 (1986), 385394.##[16] H.P. Sankappanavar, Heyting algebras with a dual lattice endomorphism, Zeitschr.##f. math. Logik und Grundlagen d. Math. 33 (1987), 565{573.##[17] H.P. Sankappanavar, SemiDe Morgan algebras, J. Symbolic. Logic 52 (1987), 712##[18] H.P. Sankappanavar, SemiHeyting algebras: An abstraction from Heyting algebras,##Actas del IX Congreso Dr. A. Monteiro (2007), 3366.##[19] H.P. Sankappanavar, SemiHeyting algebras II. In Preparation.##[20] H.P. Sankappanavar, Expansions of semiHeyting algebras. I: Discriminator varieties,##Studia Logica 98 (12) (2011), 2781.##[21] H.P. Sankappanavar, Expansions of semiHeyting algebras. II. In Preparation.##[22] J, Varlet, A regular variety of type h2; 2; 1; 1; 0; 0i, Algebra Universalis 2 (1972),##[23] H. Werner, Discriminator Algebras", Studien zur Algebra und ihre Anwendungen,##Band 6, Academie{Verlag, Berlin, 1978.##]
Dually quasiDe Morgan Stone semiHeyting algebras II. Regularity
2
2
This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, that the variety $mathbf{RDQDStSH_1}$ of regular dually quasiDe Morgan Stone semiHeyting algebras of level 1 is the join of the variety generated by the twenty 3element $mathbf{RDQDStSH_1}$chains and the variety of dually quasiDe Morgan Boolean semiHeyting algebrasthe latter is known to be generated by the expansions of the three 4element Boolean semiHeyting algebras. As consequences of our main theorem, we present (equational) axiomatizations for several subvarieties of $mathbf{RDQDStSH_1}$. The paper concludes with some open problems for further investigation.
1

65
82


Hanamantagouda P.
Sankappanavar
Department of Mathematics, State University of New York, New Paltz, NY 12561
Department of Mathematics, State University
Iran
sankapph@newpaltz.edu
Regular dually quasiDe Morgan semiHeyting algebra of level 1
dually pseudocomplemented semiHeyting algebra
De Morgan semiHeyting algebra
strongly blended dually quasiDe Morgan Stone semiHeyting algebra
discriminator variety
simple
directly indecomposable
subdirectly irreducible
equational base
[[1] R. Balbes and PH. Dwinger, Distributive Lattices", Univ. of Missouri Press,##Columbia, 1974. ##[2] S. Burris and H.P. Sankappanavar, A Course in Universal Algebra", Springer Verlag, New York, 1981. The free, corrected version (2012) is available online as a PDF file at math.uwaterloo.ca/snburris.##[3] B. Jhonsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110121.##[4] W. McCune, Prover9 and Mace 4, http://www.cs.unm.edu/mccune/prover9/.##[5] H. Rasiowa, An Algebraic Approach to NonClassical Logics", North{Holland Publ.Comp., Amsterdam, 1974.##[6] H.P. Sankappanavar, Heyting algebras with dual pseudocomplementation, Pacific J. Math. 117 (1985), 405415.##[7] H.P. Sankappanavar, Heyting algebras with a dual lattice endomorphism, Zeitschr. f. math. Logik und Grundlagen d. Math. 33 (1987), 565{573.##[8] H.P. Sankappanavar, SemiDe Morgan algebras, J. Symbolic. Logic 52 (1987), 712 724.##[9] H.P. Sankappanavar, SemiHeyting algebras: An abstraction from Heyting algebras, Actas del IX Congreso Dr. A. Monteiro (2007), 3366.##[10] H.P. Sankappanavar, Expansions of semiHeyting algebras. I: Discriminator varieties, Studia Logica 98 (12) (2011), 2781.##[11] H.P. Sankappanavar, Dually quasiDe Morgan Stone semiHeyting Regularity, Categ. General Alg. Struct. Appl. 2(1) (2014), 4764.##]
Injectivity in a category: an overview on smallness conditions
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2
Some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness. In this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by Bernhard Banaschewski and Walter Tholen, who studied these notions in a much more categorical setting. Then, we study these notions as well as the well behavior of injectivity, in the class $mod(Sigma, {mathcal E})$ of models of a set $Sigma$ of equations in a suitable category, say a Grothendieck topos ${mathcal E}$, given by M.Mehdi Ebrahimi. We close the paper by some examples to support the results.
1

83
112


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


Mahdieh
Haddadi
Department of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran.
Department of Mathematics, Statistics and
Iran
m.haddadi@semnan.ac.ir


Mojgan
Mahmoudi
Department of Mathematics, Shahid Beheshti University, G.C., Tehran 19839, Iran.
Department of Mathematics, Shahid Beheshti
Iran
mmahmoudi@sbu.ac.ir
Cogenerating set
essential extension
residual smallness
injective
[[1] J. Adamek, H. Herrlich, and G.E. Strecker, Abstract and Concrete Categories", John Wiley and Sons, Inc., 1990.##[2] F.W.. Anderson and K.R. Fuller, Rings and Categories of Modules", Springer, New York, 1974.##[3] B. Banaschewski, Injectivity and essential extensions in equational classes of algebras, Queen's Paper in Pure Appl. Math. 25 (1970), 131147.##[4] H. Barzegar and M.M. Ebrahimi, Sequentially pure monomorphisms of acts over semigroups, Eur. J. Pure Appl. Math. 1(4) (2008), 4155.##[5] H. Barzegar, M.M. Ebrahimi, and M. Mahmoudi, Essentiality and injectivity relative to sequential purity of acts, Semigroup Forum 79(1) (2009), 128144.##[6] P. Berthiaume, The injective envelope of SSets, Canad. Math. Bull. 10(2) (1967), 261273.##[7] K.R. Bhutani, Injectivity and injective hulls of abelian groups in a localic topos, Bull.##Austral. Math. Soc. 37 (1988), 4359.##[8] S. Burris and H.P. Sankapanavar, A Course in Universal Algebra", Graduate Texts in Math. No. 78, SpringerVerlag, 1981.##[9] D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, with Applications to Topology, Algebra, and Discrete Mathematics", Mathematics and its Applications, Kluwer Academic Publisher, 1995. ##[10] M. M. Ebrahimi, M. Haddadi, and M. Mahmoudi, Injectivity in a category: An overview of well behaviour theorems, Algebra, Groups, and Geometries 26 (2009), 451472.##[11] M.M. Ebrahimi, Algebra in a Grothendieck Topos: Injectivity in quasiequational classes, J. Pure Appl. Algebra 26 (1982), 269280.##[12] M.M. Ebrahimi, Equational compactness of sheaves of algebras on a Noetherian locale, Algebra Universalis 16 (1983), 318330.##[13] M.M. Ebrahimi, Internal completeness and injectivity of Boolean algebras in the topos of Msets, Bull. Austral. Math. Soc. 41(2) (1990), 323332.##[14] M.M. Ebrahimi, On ideal closure operators of Msets, Southeast Asian Bull. Math. 30 (2006), 439444.##[15] M.M. Ebrahimi and M. Mahmoudi, Purity and equational compactness of projection algebras, Appl. Categ. Structures 9(4) (2001), 381394.##[16] M.M. Ebrahimi and M. Mahmoudi, The category of Msets, Italian J. Pure Appl. Math. 9 (2001), 123132.##[17] M.M. Ebrahimi and M. Mahmoudi, Baer criterion and injectivity of projection algebras, Semigroup Forum 71(2) (2005), 332335.##[18] M.M. Ebrahimi, M. Mahmoudi, and Gh. Moghaddasi Angizan, Injective hulls of acts over left zero semigroups, Semigroup Forum 75(1) (2007), 212220.##[19] M.M. Ebrahimi, M. Mahmoudi, and Gh. Moghaddasi Angizan, On the Baer criterion for acts over semigroups, Comm. Algebra 35(12) (2007), 39123918.##[20] M.M. Ebrahimi, M. Mahmoudi, and L. Shahbaz, Proper behaviour of sequential injectivity of acts over semigroups, Comm. Algebra 37(7) (2009), 25112521.##[21] R. Goldblatt, Topoi: The Categorial Analysis of Logic", North Holland, 1986.##[22] M. Kilp, U. Knauer, and A. Mikhalev, Monoids, Acts and Categories", Walter de Gruyter, Berlin, New York, 2000.##[23] S. Maclane, Categories for theWorking Mathematicians", Graduate Texts in Mathematics, No. 5, SpringerVerlag, 1971.##[24] M. Mahmoudi, Internal injectivity of Boolean algebras in MSet, Algebra Universalis 41 (1999), 155175.##[25] M. Mahmoudi and Gh. Moghaddasi Angizan, Injective hulls of acts over idempotent semigroups, Semigroup Forum 74(2) (2007), 240246. ##[26] M. Mahmoudi and L. Shahbaz, Sequentially dense essential monomorphisms of acts over semigroups, Appl. Categ. Structures 18(5) (2010), 461471.##[27] M. Mahmoudi and L. Shahbaz, Characterizing semigroups by sequentially dense injective acts, Semigroup Forum 75(1)(2007), 116128.##[28] W. Taylor, Residually small varieties, Algebra Universalis 2 (1972), 3353. ##[29] B.R. Tennison, Sheaf Theory", Cambridge University Press, 1975.##[30] W. Tholen, Injective Objects and Cogenerating sets, J. Algebra 73(1) (1981), 139155.##[31] W. Wechiler, Universal Algebra for Computer Scientists", EATCS Monographs on Theoretical Computer Science, SpringerVerlag, 1992.##]