2020
13
1
0
0
1

Product preservation and stable units for reflections into idempotent subvarieties
https://cgasa.sbu.ac.ir/article_87414.html
10.29252/cgasa.13.1.1
1
We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semileftexact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units if and only if it is simple and the abovementioned condition holds.
0

1
22


Isabel A.
Xarez
Department of Mathematics, University of Aveiro, Portugal.
Iran
isabel.andrade@ua.pt


Joao J.
Xarez
CIDMA  Center for Research and Development in Mathematics and Applications,
Department of Mathematics, University of Aveiro, Portugal.
Iran
xarez@ua.pt
Semileftexactness
stable units
simple reflection
preservation of finite products
varieties of universal algebras
idempotent
[[1] Borceux, F. and Janelidze, G., "Galois Theories", Cambridge University Press, 2000.##[2] Carboni, A., Janelidze, G., Kelly, G.M., and Paré, R., On localization and stabilization for factorization systems, Appl. Categ. Structures 5 (1997), 158.##[3] Cassidy, C., Hébert, M., and Kelly, G.M., Reflective subcategories, localizations and factorization systems, J. Aust. Math. Soc. 38A (1985), 287329.##[4] Even, V., A Galoistheoretic approach to the covering theory of quandles, Appl. Categ. Structures 22 (2014), 817831.##[5] Grillet, P.A., "Abstract Algebra", 2nd ed., Springer, 2007.##[6] Janelidze, G., Laan, V., and Márki, L., Limit preservation properties of the greatest semilattice image functor, Internat. J. Algebra Comput. 18(5) (2008), 853867.##[7] Mac Lane, S., "Categories for the Working Mathematician", 2nd ed., Springer, 1998.##[8] Xarez, I.A., "Reflections of Universal Algebras into Semilattices, their Galois Theories and Related Factorization Systems", University of Aveiro, Ph.D. Thesis, 2013.##[9] Xarez, I.A. and Xarez, J.J., Galois theories of commutative semigroups via semilattices, Theory Appl. Categ. 28(33) (2013), 11531169.##[10] Xarez, J.J., Generalising connected components, J. Pure Appl. Algebra 216(89) (2012), 18231826.##]
1

The nonabelian tensor product of normal crossed submodules of groups
https://cgasa.sbu.ac.ir/article_87437.html
10.29252/cgasa.13.1.23
1
In this article, the notions of nonabelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.
0

23
44


Alireza
Salemkar
Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, Iran.
Iran
salemkar@sbu.ac.ir


Tahereh
Fakhr Taha
Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran 19839, Iran.
Iran
taha.tmu84@gmail.com
crossed module
tensor product
exterior product
[[1] Arias, D. and Ladra, M., The precise center of a crossed module, J. Group Theory 12 (2009), 247269.##[2] Bacon, M.R. and Kappe, L.C., The nonabelian tensor square of a 2generator pgroup of class 2, Arch. Math. (Basel) 61 (1993), 508516.##[3] Bacon, M.R., Kappe, L.C., and Morse, R.F., On the nonabelian tensor square of a 2Engel group, Arch. Math. (Basel) 69 (1997), 353364.##[4] Brown, R., Johnson, D.L., and Robertson, E.F., Some computations of nonabelian tensor products of groups, J. Algebra 111 (1987), 177202.##[5] Brown, R. and Loday, J.L., Excision homotopique en basse dimension, C.R. Acad. Sci. Ser. I Math. Paris 298 (1984), 353356.##[6] Brown, R. and Loday, J.L., Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311335.##[7] Carrasco, P., Cegarra, A.M., and Grandje'an, A.R., (Co)Homology of crossed modules, J. Pure Appl. Algebra 168 (2002), 147176.##[8] Dennis, R.K., In search of new homology functors having a close relationship to Ktheory, preprint, Cornell University, Ithaca, NY, 1976##[9] Donadze, G., Ladra, M., and Thomas, V.Z., On some closure properties of the nonabelian tensor product, J. Algebra 472 (2017), 399413.##[10] Ellis, G., The nonabelian tensor product of finite groups is finite, J. Algebra 111 (1987), 203205.##[11] Grandje'an, A.R., and Ladra, M., On totally free crossed modules, Glasgow Math. J. 40 (1998), 323332.##[12] Grandje'an, A.R. and Ladra, M., H2(T; G; $sigma$) and central extensions for crossed modules, Proc. Edinburg Math. Soc. 42 (1999), 169177.##[13] Ladra, M., and Grandje'an, A.R., Crossed modules and homology, J. Pure Appl. Algebra. 95 (1994), 4155.##[14] Miller, C., The second homology of a group, Proc. Amer. Math. Sot. 3 (1952), 588595.##[15] Mohammadzadeh, H., Shahrokhi, S., and Salemkar, A.R., Some results on stem covers of crossed modules, J. Pure Appl. Algebra 218 (2014), 19641972.##[16] Moravec, P., The nonabelian tensor product of polycyclic groups is polycyclic, J. Group Theory 10 (2007), 795798.##[17] Nakaoka, I.N., Nonabelian tensor products of solvable groups, J. Group Theory 3 (2000), 157167.##[18] Norrie, K.J., Crossed modules and analogues of group theorems, Thesis, King’s College, Univ. of London, London, 1987.##[19] Pirashvili, T., Ganea term for CCGhomology of crossed modules, Extracta Mathematicae 15 (2000), 231235.##[20] Salemkar, A.R., Talebtash, S., and Riyahi, Z., The nilpotent multipliers of crossed modules, J. Pure Appl. Algebra 221 (2017), 21192131.##[21] Vieites, A.M. and Casas, J.M., Some results on central extensions of crossed modules, Homology, Homotopy Appl. 4 (2002), 2942.##[22] Visscher, M.P., On the nilpotency class and solvability length of nonabelian tensor products of groups, Arch. Math. 73 (1999), 161171.##[23] Whitehead, J.H.C., On adding relations to homotopy groups, Ann. of Math 42 (1942), 409428.##]
1

Distributive lattices with strong endomorphism kernel property as direct sums
https://cgasa.sbu.ac.ir/article_87512.html
10.29252/cgasa.13.1.45
1
Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem 2.8}). We shall determine the structure of special elements (which are introduced after Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of three lattices, a lattice with exactly one strong element, a lattice which is a direct sum of 2 element lattices with distinguished elements 1 and a lattice which is a direct sum of 2 element lattices with distinguished elements 0, and the sublattice of strong elements is isomorphic to a product of last two mentioned lattices.
0

45
54


Jaroslav
Gurican
Department of Algebra and Geometry,
Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Slovakia.
Iran
gurican@fmph.uniba.sk
unbounded distributive lattice
strong endomorphism kernel property
congruence relation
bounded Priestley space
Priestley duality
strong element
direct sum
[[1] Blyth, T.S., Fang, J., and Silva, H.J., The endomorphism kernel property in finite distributive lattices and de Morgan algebras, Comm. Algebra 32(6) (2004), 2225 2242.##[2] Blyth, T.S., Fang, J., and Wang, L.B., The strong endomorphism kernel property in distributive double palgebras, Sci. Math. Jpn. 76(2) (2013), 227234.##[3] Blyth, T.S. and Silva, H.J., The strong endomorphism kernel property in Ockham algebras, Comm. Algebra 36(5) (2008), 16821694.##[4] Clark, D.M. and Davey, B.A., Natural Dualities for the Working Algebraist", Cambridge University Press, 1998.##[5] Davey, B.A. and Priestley, H.A., Introduction to Lattices and Order", 2nd edn. Cambridge University Press, 2002.##[6] Fang, G. and Fang, J., The strong endomorphism kernel property in distributive palgebras, Southeast Asian Bull. Math. 37(4) (2013), 491497.##[7] Fang, J. and Sun, Z.J., Semilattices with the strong endomorphism kernel property, Algebra Universalis 70(4) (2013), 393401.##[8] Fang, J., The Strong endomorphism kernel property in double MSalgebras, Studia Logica 105(5) (2017), 9951013.##[9] Gratzer, G., Lattice theory: Foundation", Birkhauser, 2011.##[10] Gurican, J., Strong endomorphism kernel property for Brouwerian algebras, JP J. Algebra Number Theory Appl. 36(3) (2015), 241258.##[11] Gurican, J. and Ploscica M., The strong endomorphism kernel property for modular palgebras and distributive lattices, Algebra Universalis 75(2) (2016), 243255.##[12] Haluskova, E., Strong endomofphism kernel property for monounary algebras, Math. Bohem. 143(2) (2018), 161171.##[13] Kaarli, K. and Pixley, A.F., Polynomial completeness in algebraic system", Chapman & Hall/CRC, 2001.##[14] Ploscica, M., Afine completions of distributive lattices, Order 13(3) (1996), 295311.##]
1

Separated finitely supported $Cb$sets
https://cgasa.sbu.ac.ir/article_87413.html
10.29252/cgasa.13.1.55
1
The monoid $Cb$ of name substitutions and the notion of finitely supported $Cb$sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported $Cb$set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported $Cb$set is an injective map. Also, for every two distinct elements of a simple finitely supported $Cb$set, there exists an element of the monoid $Cb$ which separates them by making just one of them into an element with the empty support.In this paper, we generalize these properties of simple finitely supported $Cb$sets by modifying slightly the notion of the support map; defining the notion of $mathsf{2}$equivariant support map; and introducing the notions of sseparated and zseparated finitely supported $Cb$sets. We show that the notions of sseparated and zseparated coincide for a finitely supported $Cb$set whose support map is $mathsf{2}$equivariant. Among other results, we find a characterization of simple sseparated (or zseparated) finitely supported $Cb$sets. Finally, we show that some subcategories of finitely supported $Cb$sets with injective equivariant maps which constructed applying the defined notions are reflective.
0

55
82


Khadijeh
Keshvardoost
Department of Mathematics, Velayat University, Iranshahr, Sistan and Baluchestan, Iran.
Iran
hkeshvardoust@yahoo.com


Mojgan
Mahmoudi
Department of Mathematics,
Shahid Beheshti University, Tehran 19839, Iran.
Iran
mmahmoudi@sbu.ac.ir
Finitely supported $Cb$sets
nominal set
$S$set
support
simple
[[1] Adamek, J., Herrlich, H., and Strecker, G.E., "Abstract and Concrete Categories", John Wiley and Sons, 1990.##[2] Burris, S. and Sankappanavar, H.P., "A Course in Universal Algebra", Springer Velag, 1981.##[3] Ebrahimi, M.M., Keshvardoost, Kh., and Mahmoudi, M., Simple and subdirectly irreducible finitely supported Cbsets, Theort. Comput. Sci. 706 (2018), 121.##[4] Ebrahimi, M.M. and Mahmoudi, M., The category of Msets, Ital. J. Pure Appl. Math. 9 (2001), 123132.##[5] Gabbay, M. and Pitts, A., A new approach to abstract syntax with variable binding, Form. Asp. Comput. 13(35) (2002), 341363.##[6] Herrlich, H. and Strecker, G., Coreflective subcategories, Trans. Amer. Math. Soc. 157 (1971), 205226.##[7] Kilp, M., Knauer, U. and Mikhalev, A., "Monoids, Acts and Categories", Walter de Gruyter, 2000.##[8] Pitts, A., "Nominal sets, Names and Symmetry in Computer Science", Cambridge University Press, 2013.##[9] Pitts, A., Nominal presentations of the cubical sets model of type theory, LIPIcs. Leibniz Int. Proc. Inform. (2015), 202220.##]
1

A classification of hull operators in archimedean latticeordered groups with unit
https://cgasa.sbu.ac.ir/article_87552.html
10.29252/cgasa.13.1.83
1
The category, or class of algebras, in the title is denoted by $bf W$. A hull operator (ho) in $bf W$ is a reflection in the category consisting of $bf W$ objects with only essential embeddings as morphisms. The proper class of all of these is $bf hoW$. The bounded monocoreflection in $bf W$ is denoted $B$. We classify the ho's by their interaction with $B$ as follows. A ``word'' is a function $w: {bf hoW} longrightarrow {bf W}^{bf W}$ obtained as a finite composition of $B$ and $x$ a variable ranging in $bf hoW$. The set of these,``Word'', is in a natural way a partially ordered semigroup of size $6$, order isomorphic to ${rm F}(2)$, the free $01$ distributive lattice on $2$ generators. Then, $bf hoW$ is partitioned into $6$ disjoint pieces, by equations and inequations in words, and each piece is represented by a characteristic orderpreserving quotient of Word ($approx {rm F}(2)$). Of the $6$: $1$ is of size $geq 2$, $1$ is at least infinite, $2$ are each proper classes, and of these $4$, all quotients are chains; another $1$ is a proper class with unknown quotients; the remaining $1$ is not known to be nonempty and its quotients would not be chains.
0

83
104


Ricardo E.
Carrera
Department of Mathematics, Nova Southeastern University, 3301 College Ave., Fort Lauderdale, FL, 33314, USA.
Iran


Anthony W.
Hager
Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.
Iran
ahager@wesleyan.edu
latticeordered group
Archimedean
weak unit
bounded monocoreflection
essential extension
hull operator
partially ordered semigroup
[[1] Anderson, M.E. and Feil, T.H., "Latticeordered Groups: an Introduction", Springer Science & Business Media, 2012.##[2] Balbes, R. and Dwinger, P., "Distributive lattices", New York, 1974.##[3] Ball, R. and Hager, A.W., Algebraic extensions of an archimedean latticeordered group I, J. Pure Appl. Algebra 85(1) (1993), 120.##[4] Ball, R., Hager, A.W., and Neville, C., The QuasiFK cover of Compact Hausdorff Space and the ACIdeal Completion of an Archimedean lGroup, General Top. Appl.: Proceedings of the 1988 Northeast Conference 123, 1990.##[5] Banaschewski, B. and Hager, A., Representation of Hclosed monoreflections in archimedean lgroups with weak unit, Categ. General Alg. Struct. Appl. 9(1) (2018), 113.##[6] Bigard, A., Keimel, K., and Wolfenstein, S., "Groupes et Anneaux R´eticul´es", Springer, 2006.##[7] Birkhoff, G., "Lattice theory", American Mathematical Society, 1940.##[8] Carrera, R., Various Completeness of an Archimedean Latticeordered Group, 2014.##[9] Carrera, R. and Hager, A.W., On hull classes of lgroups and covering classes of spaces, Math. Slovaca 61(3) (2011), 411428.##[10] Carrera, R.E. and Hager, A.W., Bsaturated hull classes in `groups and covering classes of spaces, Appl. Categ. Structures, 23(5) (2015), 709723.##[11] Carrera, R.E. and Hager, A.W., Bounded equivalence of hull classes in archimedean latticeordered groups with unit, Appl. Categ. Structures 24(2) (2016), 163179.##[12] Conrad, P., The essential closure of an archimedean latticeordered group, Duke Math. J. 38(1) (1971), 151160.##[13] Conrad, P., The hulls of representable lgroups and frings, J. Austral. Math. Soc. 16(4) (1973), 385415.##[14] Darnel M. R., "The Theory of LatticeOrdered Groups", Pure Appl. Math. 187, Marcel Dekker, 1995.##[15] Engelking, R."Outline of General Topology", NorthHolland Pub. Co., 1968.##[16] Fine, N., Gillman, L., and Lambek, J., Rings of Quotients of Rings of Functions, McGill University Press, 1996.##[17] Fuchs, L., "Partially Ordered Algebraic Systems", Courier Corporation, 2011.##[18] Gillman, L. and Jerison, M., "Rings of Continuous Functions", Courier Dover Publications, 2017.##[19] Gleason, A., Projective topological spaces, Illinois J. Math. 2(4A) (1958), 482489.##[20] Hager, A.W., Algebraic closures of lgroups of continuous functions, in "Rings of Continuous Functions", C. Aull, Ed., Lecture Notes Pure Appl. Math. 95 (1985), Marcel Dekker, 165194.##[21] Hager, A.W., Minimal covers of topological spaces, Ann. New York Acad. Sci. 552(1) (1989), 4459.##[22] Hager, A.W. and Martinez, J.,"$alpha$projectable and laterally $alpha$complete archimedean latticeordered groups", in S. Bernahu (ed.), Proc. Conf. on Mem. of T. Retta, Temple U., PA/Addis Ababa (1995), Ethiopian J. Sci., 73–84.##[23] Hager, A.W. and Robertson, L., Representing and ringifying a Riesz space, Sympos. Math 21 (1977), 411431.##[24] Hager, A.W. and Robertson, L., Extremal units in an archimedean Riesz space, Rendiconti del Seminario matematico della Universit`a di Padova 59 (1978), 97115.##[25] Henriksen, M., Vermeer, J., and Woods, R., Wallman covers of compact spaces, Instytut Matematyczny Polskiej Akademi Nauk (Warszawa), 1989.##[26] Herrlich, H. and Strecker, G., "Category Theory: an Introduction", Heldermann Verlag, 1979.##[27] Martinez, J., Hull classses of archimedean latticeordered groups with unit: A survey", in "Ordered Algebraic Structures", Springer, 2002, 89121.##[28] Martinez, J., Polar functions—II: Completion classes of archimedean falgebras vs. covers of compact spaces, J. Pure Appl. Algebra 190(13) (2004), 225249.##[29] Porter, R. and Woods G., "Extensions and absolutes of Hausdorff spaces", Springer Science & Business Media, 2012.##[30] Woods, G., Covering Properties and Coreflective Subcategories a, b, Ann. New York Acad. Sci. 552(1) (1989), 173184.##]
1

The symmetric monoidal closed category of cpo $M$sets
https://cgasa.sbu.ac.ir/article_87434.html
10.29252/cgasa.13.1.105
1
In this paper, we show that the category of directed complete posets with bottom elements (cpos) endowed with an action of a monoid $M$ on them forms a monoidal category. It is also proved that this category is symmetric closed.
0

105
124


Halimeh
Moghbeli
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran
Iran
h_moghbeli@ujiroft.ac.ir
Directed complete partially ordered set
$M$sets
symmetric monoidal closed category
[[1] Abramsky, S. and Jung, A., "Domain Theory", Handbook of logic in computer science (Vol. 3). Oxford University Press, 1995.##[2] Borceux, F., "Handbook of Categorical Algebra 1: Basic Category Theory", Cambridge University Press, Cambridge, 1994.##[3] Borceux, F., "Handbook of Categorical Algebra 2: Categories and Structures", Cambridge University Press, Cambridge, 1994.##[4] Davey, B.A. and Priestly, H.A., "Introduction to Lattices and Order", Cambridge University Press, Cambridge, 1990.##[5] Day, B.J., On closed categories of functors, Reports of the midwest category seminar (Lane, S.Mac, editor), Lecture Notes in Math., SpringerVerlag, BerlinNew York, 137 (1970), 1–38.##[6] Ebrahimi, M.M. and Mahmoudi, M., The category of MSets, Ital. J. Pure Appl. Math. 9 (2001), 123132.##[7] Fiech, A., Colimits in the category Dcpo, Math. Structures Comput. Sci., 6 (1996), 455468.##[8] Jung, A., "Cartesian closed categories of Domain", Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.##[9] Kilp, M., Knauer, U., and A. Mikhalev, "Monoids, Acts and Categories", Walter de Gruyter, Berlin, New York, 2000.##[10] Mahmoudi, M. and Moghbeli, H., The category of Sacts in the category Cpo, Bull. Iran. Math. Soc. 41(1) (2015), 159175.##[11] Mahmoudi, M. and Moghbeli, H., The categories of actions of a dcpomonoid on directed complete posets, Quaigroups Relatd Sytems, 23 (2015), 283295.##[12] MoghbeliDamaneh, H., Actions of a separately cpomonoid on pointed directed complete posets, Categ. General Alg. Struct. Appl., 3(1) (2015), 2142.##[13] Mac Lane, S., "Categories for the working mathematician". Vol.5. Springer Science and Business Media, 2013.##[14] Plotkin, G.D., A powerdomain construction. SIAM Journal on Computing, 5 (1976), 452487.##[15] Plotkin, G.D., A powerdomain for countable nondeterminism. In M. Nielsen and E. M. Schmidt, editors, Automata, Languages and programming, volume 140 of Lecture Notes in Computer Science, pages 412428. EATCS, Springer Verlage, 1982.##[16] Smyth, M.B., Powerdomains. Journal of Computer and Systems Sciences, 16 (1978), 2336.##[17] Streicher, T., "Domaintheoretic Foundations of Functional Programming". World Scientific, Singapore, 2006.##[18] Tix, R., Keimel. K., and G. D. Plotkin, "Semantic Domains for Combining Probability and NonDeterminism", Electronic Notes in Theoretical Computer Science, 222 (2009), 399.##]
1

Crossed squares, crossed modules over groupoids and cat$^{bf {12}}$groupoids
https://cgasa.sbu.ac.ir/article_87511.html
10.29252/cgasa.13.1.125
1
The aim of this paper is to introduce the notion of cat$^{bf {1}}$groupoids which are the groupoid version of cat$^{bf {1}}$groups and to prove the categorical equivalence between crossed modules over groupoids and cat$^{bf {1}}$groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat$^{bf {2}}$groupoids, and then we show their categories are equivalent. These equivalences enable us to obtain more examples of groupoids.
0

125
142


Sedat
Temel
Department of Mathematics, Faculty of Arts and Science, Recep Tayyip Erdogan University, Rize, Turkey.
Iran
sedat.temel@erdogan.edu.tr
crossed module
crossed square
groupoid
cat$^{bf {1}}$group
cat$^{bf {2}}$group
[[1] Akiz, H.F., Alemdar, N., Mucuk, O., Sahan, T., Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Math. J. 20(2) (2013), 223238.##[2] Ataseven, C ., Relations among higher order crossed modules over groupoids, Konu ralp J. Math. 4(1) (2016), 282290.##[3] Baez, J.C. and Lauda, A.D., Higher dimensional algebra V: 2groups, Theory Appl. Categ. 12(14) (2004), 423491.##[4] Brown, R., "Topology and Groupoids", BookSurge LLC, 2006.##[5] Brown, R. and Spencer, C.B., Ggroupoids, crossed modules and the fundamental groupoid of a topological group, Indag. Math. (N.S.) 79(4) (1976), 296302.##[6] Brown, R., Higgins, P.J., Sivera, R., "Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids", Eur. Math. Soc. Tracts in Math. 15, 2011.##[7] Brown, R. and Higgins, P.J., Tensor products and homotopies for !groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987), 133.##[8] Brown R. and Higgins P.J., Crossed complexes and nonabelian extensions, In: Cat egory Theory. Lecture Notes in Math. 962, Springer, 1982.##[9] Brown, R. and Icen, I., Homotopies and Automorphisms of Crossed Module Over Groupoids, Appl. Categ. Structures 11 (2003), 185206.##[10] Brown, R. and Loday, J.L., Van Kampen theorems for diagrams of spaces, J. Topol. 26(3) (1987), 311335.##[11] Ellis, G. and Steiner, R., Higherdimensional crossed modules and the homotopy groups of (n+1)ads, J. Pure Appl. Algebra 46 (1987), 117136.##[12] Gilbert, N.D., Derivations, automorphisms and crossed modules, Comm. Algebra 18(8) (1990), 27032734.##[13] GuinWalery, D. and Loday, J.L., Obstruction a l'excision en Ktheorie algebrique, In: Algebraic Ktheory, Lecture Notes in Math. 854, Springer, 1981.##[14] Huebschmann, J., Crossed nfold extensions of groups and cohomology, Comment. Math. Helv. 55 (1980), 302314.##[15] Icen, I., The equivalence of 2groupoids and crossed modules, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 49 (2000), 3948.##[16] Loday, J.L., Cohomologie et groupe de Steinberg relatifs, J. Algebra 54 (1978), 178 202.##[17] Loday, J.L., Spaces with afnitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24(2) (1982), 179202.##[18] Mackenzie, K., "Lie Groupoids and Lie Algebroids in Differential Geometry", Cam bridge University Press, 1987.##[19] Maclane, S., "Categories for the Working Mathematician", Springer, 1971.##[20] Mucuk, O. and Demir, S., Normality and quotient in crossed modules over groupoids and double groupoids, Turkish J. Math. 42 (2018), 23362347.##[21] Mucuk, O. and Sahan, T., Groupgroupoid actions and liftings of crossed modules, Georgian Math. J. 26(3) (2019), 437447.##[22] Mucuk, O. and Sahan, T., Alemdar, N., Normality and quotients in crossed modules and groupgroupoids, Appl. Categ. Structures 23(3) (2015), 415428.##[23] Norrie, K., Actions and automorphisms of crossed modules, Bull. Soc. Math. France 118 (1990), 129146.##[24] Temel, S., Normality and quotient in crossed modules over groupoids and 2 groupoids, Korean J. Math. 27(1) (2019), 151163.##[25] Whitehead, J.H.C., Combinatorial homotopy II, Bull. Amer. Math. Soc. (N.S.) 55 (1949), 453496.##[26] Whitehead, J.H.C., Note on a previous paper entitled "On adding relations to ho motopy groups", Ann. of Math. (2) 47 (1946), 806810.##]
1

Tense like equality algebras
https://cgasa.sbu.ac.ir/article_87465.html
10.29252/cgasa.13.1.143
1
In this paper, first we define the notion of involutive operator on bounded involutive equality algebras and by using it, we introduce a new class of equality algebras that we called it a tense like equality algebra. Then we investigate some properties of tense like equality algebra. For two involutive bounded equality algebras and an equality homomorphism between them, we prove that the tense like equality algebra structure can be transfer by this equality homomorphism. Specially, by using a bounded involutive equality algebra and quotient structure of it, we construct a quotient tense like equality algebra. Finally, we investigate the relation between tense like equality algebras and tense MValgebras.
0

143
166


Mohammad Ali
Hashemi
Department of Mathematics, Payame Noor University, P.O.Box 193953697, Tehran, Iran.
Iran
ali1978hashemi@gmail.com


Rajabali
Borzooei
Department of Mathematics, Shahid Beheshti University, Tehran, Iran.
Iran
borzooei@sbu.ac.ir
Equality algebra
tense like equality algebra
MValgebra
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