On injective constructions of S-semigroups. Jan Paseka Masaryk University
|
|
- Melinda Lyons
- 5 years ago
- Views:
Transcription
1 On injective constructions of S-semigroups Jan Paseka Masaryk University Joint work with Xia Zhang South China Normal University BLAST 2018 University of Denver, Denver, USA Jan Paseka (MU) / 39
2 Contents 1 Introduction M Motivation S Sierpiński space and injectivity B Complete Boolean algebras and injectivity Q Backgrounds: Quantales and quantale-like structures H Backgrounds: Injective hulls for some partially ordered algebras 2 Injective hulls of posemigroups O Injective objects of posemigroups C Construction of a special closure operator H Injective hulls of posemigroups 3 Injective hulls for S-semigroups O Injective objects of S-semigroups C Construction of a special closure operator H Injective hulls of S-semigroups 4 References Jan Paseka (MU) / 39
3 Introduction: Motivation Fuzzy logic It is well-known that the semantics of a given fuzzy logic can be formally axiomatized by means of a residuated poset. The appropriateness of such approach is emphasized by the fact that t-norm based logic usually refers to residuated systems of fuzzy logic with t-norm based semantics. Here the conjunction connective is interpreted by a t-norm and the implication operator by its residuum. In essence, this fact is the source of numerous examples of prospective truth functions of connectives for fuzzy logic. Hence, we assume that a corresponding residuated poset is given and its connection to the semantics of a fuzzy logic is known. Content of the work In this paper, we shall investigate the injective hulls in the category of S-semigroups over a posemigroup S and we obtain their concrete form. Jan Paseka (MU) / 39
4 1. Introduction: Sierpiński space and injectivity We shall write S for the Sierpinski space, i.e., the set 2 = {0, 1} equipped with topology Ω(S) = {S, {1}, }. Statement (i) Any T 0 -space may be embedded as a subspace of a power of S. (ii) S is an injective T 0 -space, i.e., if i : X X is an inclusion of T 0 -spaces, then every continuous map f : X S is the restriction of some continuous map g : X S. Hence, g i = f. X f i X S... (iii) A T 0 -space is injective iff it is a retract of a power of S. g Jan Paseka (MU) / 39
5 1. Introduction: Complete Boolean algebras and injectivity Statement (i) The injective objects in the category Bool of Boolean algebras are precisely the complete Boolean algebras. (ii) The injective objects in the category DLat of distributive lattices are precisely the complete Boolean algebras. C f i D B... g Jan Paseka (MU) / 39
6 1. Introduction: Quantales and quantale-like structures Definitions Definition 1 (Mulvey C.J., 1986) A structure (Q,, ) is called a quantale if (Q, ) is a -semilattice, (Q, ) is a semigroup, and multiplication distributes over arbitrary joins in both coordinates, that is, ( ) a M = {a m m M}, ( ) M a = {m a m M}, for any a Q, M Q. A quantale is commutative if the binary operation is commutative. Jan Paseka (MU) / 39
7 1. Introduction: Quantales and quantale-like structures Definitions Each quantale Q is residuated in the following natural way: s a b s a r b a s l b, a, b, s Q. Jan Paseka (MU) / 39
8 1. Introduction: Quantales and quantale-like structures Definitions A frame (locale) L is a complete lattice such that for any a L, M L. a ( M) = {a m m M}, Jan Paseka (MU) / 39
9 1. Introduction: Quantale modules Definitions Definition 2 (Cf. [6, 11]) Given a quantale Q, a left quantale module is a pair (A, ), where A is a -lattice and : Q A A such that: 1 q ( S) = s S (q s) for every q Q, S A; 2 ( T ) a = t T (t a) for every a A, T Q; 3 q 1 (q 2 a) = (q 1 q 2 ) a for every a A, q 1, q 2 Q. Q-modules are equivalent to so called Q-sup-lattices, where Q is a unital (commutative) quantale. This fact was first pointed out by Stubbe in 2006 and later proved in quantale-like setting by Solovyov in Thus, Q-modules can be seen as a fuzzification of complete lattices. Jan Paseka (MU) / 39
10 1. Introduction: Quantale modules Definitions Definition 2 (Cf. [6, 11]) Given a quantale Q, a left quantale module is a pair (A, ), where A is a -lattice and : Q A A such that: 1 q ( S) = s S (q s) for every q Q, S A; 2 ( T ) a = t T (t a) for every a A, T Q; 3 q 1 (q 2 a) = (q 1 q 2 ) a for every a A, q 1, q 2 Q. Q-modules are equivalent to so called Q-sup-lattices, where Q is a unital (commutative) quantale. This fact was first pointed out by Stubbe in 2006 and later proved in quantale-like setting by Solovyov in Thus, Q-modules can be seen as a fuzzification of complete lattices. Jan Paseka (MU) / 39
11 1. Introduction: Quantale algebras Definitions Definition 3 (Cf. [11], Solovyov) Given a commutative quantale Q, a quantale algebra is a quantale module (A, ) such that: 1 (A,, ) is a quantale; 2 q (a b) = (q a) b for every a, b A, q Q. Q-algebras are equivalent to so called Q-quantales, where Q is a unital (commutative) quantale. Thus, Q-algebras can be seen as a fuzzification of quantales. Jan Paseka (MU) / 39
12 1. Introduction: Quantale algebras Definitions Definition 3 (Cf. [11], Solovyov) Given a commutative quantale Q, a quantale algebra is a quantale module (A, ) such that: 1 (A,, ) is a quantale; 2 q (a b) = (q a) b for every a, b A, q Q. Q-algebras are equivalent to so called Q-quantales, where Q is a unital (commutative) quantale. Thus, Q-algebras can be seen as a fuzzification of quantales. Jan Paseka (MU) / 39
13 1. Introduction: Injective objects in categories Definitions Definition 4 Let C be a category and let M be a class of morphisms in C. We recall that an object S from C is M-injective in C provided that for any morphism h : A B in M and any morphism f : A S in C there exists a morphism g : B S in C such that gh = f. A f h M B... S g Jan Paseka (MU) / 39
14 1. Introduction: M-injective hull Definitions Definition 5 M-essential morphism A morphism η : A B in M is called M-essential if every morphism ψ : B C in C, for which the composite ψη is in M, is itself in M. M-injective hull An object H C is called an M-injective hull of an object S if H is M-injective and there exists an M-essential morphism η : S H. M-injective hulls are unique up to isomorphism. S f η M H... I g M Jan Paseka (MU) / 39
15 1. Introduction: M-injective hull Definitions Definition 5 M-essential morphism A morphism η : A B in M is called M-essential if every morphism ψ : B C in C, for which the composite ψη is in M, is itself in M. M-injective hull An object H C is called an M-injective hull of an object S if H is M-injective and there exists an M-essential morphism η : S H. M-injective hulls are unique up to isomorphism. S f η M H... I g M Jan Paseka (MU) / 39
16 1. Introduction: Results on injective hulls of posets 1967, Banaschewski B., Bruns G., Categorical construction of the MacNeille completion, Arch. Math. Theorem 1 (Banaschewski B., Bruns G.) For a partially ordered set P, the following conditions are equivalent: 1 P is a complete lattice; 2 P is injective in posets with respect to the class of order embeddings; 3 P is a retract of every extension, i.e., for any order embedding j : P R there is an order-preserving map k : R P such that k j = id P ; 4 P has no essential extensions, i.e., for any order embedding j : P R the poset R is isomorphic to P. Jan Paseka (MU) / 39
17 1. Introduction: Results on injective hulls of posets Theorem 2 (Banaschewski B., Bruns G.) For a partially ordered set P, the following conditions are equivalent: 1 E is a MacNeille completion of P; 2 E is an injective hull of P; 3 E is an injective extension of P not containing any properly smaller such extension of P; 4 E is an essential extension of P not contained in any properly larger such extension of P. Jan Paseka (MU) / 39
18 1. Introduction: Results on injective hulls of semilattices 1970, Bruns G., Lakser H., Injective hulls of meet-semilattices, Canadian Mathematical Bulletin Theorem 3 (Bruns G., Lakser H.) A meet-semilattice S is injective iff it is a frame, i.e., it is complete and satisfies a M = ( a m m M ), (1) for all a S, M S. Distributive joins We say that a subset M of a meet-semilattice S has a distributive join if (i) its supremum exists, and (ii) for all a S we have a M = ( a m m M ). Jan Paseka (MU) / 39
19 1. Introduction: Results on injective hulls of semilattices 1970, Bruns G., Lakser H., Injective hulls of meet-semilattices, Canadian Mathematical Bulletin Theorem 3 (Bruns G., Lakser H.) A meet-semilattice S is injective iff it is a frame, i.e., it is complete and satisfies a M = ( a m m M ), (1) for all a S, M S. Distributive joins We say that a subset M of a meet-semilattice S has a distributive join if (i) its supremum exists, and (ii) for all a S we have a M = ( a m m M ). Jan Paseka (MU) / 39
20 1. Introduction: Results on injective hulls of semilattices 1970, Bruns G., Lakser H., Injective hulls of semilattices. Bulletin I D (S) = {A S A = A ; M A has a distributive join M A}. Theorem 4 (Bruns G., Lakser H.) The injective hull of a meet-semilattice S is (up to isomorphism) I D (S). Jan Paseka (MU) / 39
21 1. Introduction: Results on injective hulls of semilattices 1970, Bruns G., Lakser H., Injective hulls of semilattices. Bulletin I D (S) = {A S A = A ; M A has a distributive join M A}. Theorem 4 (Bruns G., Lakser H.) The injective hull of a meet-semilattice S is (up to isomorphism) I D (S). Jan Paseka (MU) / 39
22 1. Introduction: Results on injective hulls of certain S-systems over a semilattice 1972, Johnson C.S., J.R., McMorris F.R., Injective hulls of certain S-systems over a semilattice. Proc. Amer. Math. Soc. Theorem 5 (Johnson C.S., J.R., McMorris F.R.) The injective hull of an S-system M S is (up to isomorphism) I D (M S ). Jan Paseka (MU) / 39
23 1. Introduction: backgrounds and motivations 1974, Schein B.M. Injectives in certain classes of semigroups. Semigroup Forum. Theorem 6. (1974 Schein) The category of semigroups has only trivial injectives. Jan Paseka (MU) / 39
24 1. Introduction: Results on injective hulls of posemigroups 2012, Lambek J., Barr M., Kennison J.F. and Raphael R., Injective hulls of partially ordered monoids. Theory Appl. Categ. The category of po-monoids Partially ordered monoids (po-monoid) with submultiplicative order-preserving mappings, i.e., an order-preserving mapping φ : A B of po-monoids satisfying φ(1) = 1, for all a, b A. φ(a) φ(b) φ(a b), Jan Paseka (MU) / 39
25 1. Introduction: Results on injective hulls of posemigroups 2012, Lambek J., Barr M., Kennison J.F. and Raphael R., Injective hulls of partially ordered monoids. Theory Appl. Categ. Theorem 7 (2012 Lambek, Barr, Kennison and Raphael) A po-monoid (S, ) is injective iff it is a quantale, i.e., it is complete and satisfies a M = ( a m m M ), (2) for all a S, M S.. Jan Paseka (MU) / 39
26 2. Injective constructions for posemigroups (Definitions) Category Pos A posemigroup (S,, ) is both a semigroup (S,, ) and a poset (S, ) such that for any a, b, c, d S, a b, c d = a c b d. Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., for all a 1, a 2 S. f(a 1 ) f(a 2 ) f(a 1 a 2 ) We denote the category of posemigroups with subhomomorphisms as morphisms by Pos. Jan Paseka (MU) / 39
27 2. Injective constructions for posemigroups (Definitions) Category Pos A posemigroup (S,, ) is both a semigroup (S,, ) and a poset (S, ) such that for any a, b, c, d S, a b, c d = a c b d. Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., for all a 1, a 2 S. f(a 1 ) f(a 2 ) f(a 1 a 2 ) We denote the category of posemigroups with subhomomorphisms as morphisms by Pos. Jan Paseka (MU) / 39
28 2. Injective constructions for posemigroups (Definitions) Category Pos A posemigroup (S,, ) is both a semigroup (S,, ) and a poset (S, ) such that for any a, b, c, d S, a b, c d = a c b d. Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., for all a 1, a 2 S. f(a 1 ) f(a 2 ) f(a 1 a 2 ) We denote the category of posemigroups with subhomomorphisms as morphisms by Pos. Jan Paseka (MU) / 39
29 2. Injective constructions for posemigroups (Definitions) Embeddings in Pos Let ε be the class of morphisms e : S T in the category Pos which satisfy the following condition: for all a 1, a 2,..., a n S. e(a 1 )... e(a n ) e(a) = a 1... a n a, Jan Paseka (MU) / 39
30 2. Injective constructions for posemigroups (Examples) An example (Rosenthal) Let (S,, ) be a posemigroup, P(S) the set of all of the downsets in S. Then (P(S),, ) is a posemigroup, where X Y = (X Y ) = {a S a x y, for some x X, y Y }, for all X, Y S. Moreover, P(A) is a complete lattice whose joins are unions. Consequently, we obtain that (P(S),, ) is a quantale. Jan Paseka (MU) / 39
31 2. Injective constructions for posemigroups (Injectives) Theorem 8 (2014 Zhang, Laan) For a posemigroup S, the following statements are equivalent: 1 S is ε -injective in Pos, 2 S is a quantale. Jan Paseka (MU) / 39
32 2. Injective constructions for posemigroups (Constructions) Definition: Nucleus A closure operator j on a posemigroup S is called a nucleus if it is a subhomomorphism. We denote S j = {a S a = j(a)}. Proposition Let (S,, ) be a posemigroup, j a nucleus on it. Then (S j, j, ) is again a posemigroup equipped with the multiplication and induced order as a j b = j(a b), for any a, b S. In addition, if S is a quantale, then (S j, j, ) is a quantale as well. Jan Paseka (MU) / 39
33 2. Injective hulls for posemigroups (Constructions) the nucleus cl: Zhang, Laan 2014, Xia, Zhao, Han 2017 Let (S,, ) be a posemigroup, D P(S). Define cl(d) := {x S ( a, c S 1, b S) a D c b = a x c b}, where S 1 is the monoid obtained from the semigroup S by externally adjoining the identity element 1. Then cl : P(S) P(S) is a nucleus on P(S) satisfying cl(x ) = x, x S. Jan Paseka (MU) / 39
34 2. Injective hulls for posemigroups Theorem Theorem 9 (Zhang, Laan 2014, Xia, Zhao, Han 2017) Let (S,, ) be a posemigroup, cl: P(S) P(S) be defined as above. Then (P(S) cl, cl, ) is the ε -injective hull of the posemigroup S. Jan Paseka (MU) / 39
35 3. Injective constructions for S-semigroups (Definitions) (S, ) is always a posemigroup A posemigroup (A,, ) together with a mapping S A A (under which a pair (s, a) maps to an element of A denoted by s a) is called an S-semigroup, denoted by S A, or simple A, if for any a, b A, s, t S, 1 s (a b) = (s a) b = a (s b), 2 (s t) a = s (t a), fulfilling a b, s t = s a t b. Jan Paseka (MU) / 39
36 3. Injective constructions for S-semigroups (Definitions) Morphisms in Ssgr An order-preserving mapping f : S A S B of S-semigroups is called a subhomomorphism if it is both submultiplicative in posemigroups, i.e., f(a 1 ) f(a 2 ) f(a 1 a 2 ) for all a 1, a 2 A, and S-submultiplicative in S-posets, i.e., s f(a) f(s a) for all a A, s S. We denote the category of S-semigroups with subhomomorphisms as morphisms by Ssgr. Jan Paseka (MU) / 39
37 3. Injective constructions for S-semigroups (Definitions) Embeddings in Ssgr Let ε 0 be the class of morphisms e : SA S B in the category Ssgr which satisfy the following conditions: and s (e(a 1 )... e(a n )) e(a) = s (a 1... a n ) a, for all a 1, a 2,..., a n, a A, s S. e(a 1 )... e(a n ) e(a) = a 1... a n a, Then ε ε 0, where ε is the class of all S-semigroup homomorphisms that are order-embeddings. Jan Paseka (MU) / 39
38 3. Injective constructions for S-semigroups (Definitions) A new quantale-like structure An S-semigroup quantale is an S-semigroup ( S A,, ) such that 1 (A,, ) is a quantale; 2 s M = {s m m M}, for every s S, M A. Jan Paseka (MU) / 39
39 3. Injective constructions for S-semigroups An example Let ( S A,, ) be an S-semigroup, P(A) the set of all of the downsets in A. Then ( S P(A),,, ) is an S-semigroup, where X Y = (X Y ) = {a A a x y, for some x X, y Y }, s X = (s X ) = {a A a s x, for some x X }, for all X, Y A, s S. Moreover, S P(A) is a complete lattice whose joins are unions. Consequently, we obtain that ( S P(A),,, ) is an S-semigroup quantale. Jan Paseka (MU) / 39
40 3. Injective constructions for S-semigroups (Injectives) Theorem 8 (2017 Zhang, Paseka) For an S-semigroup S A, the following statements are equivalent: SA is ε 0 -injective in Ssgr, SA is ε-injective in Ssgr, S A is an S-semigroup quantale. Jan Paseka (MU) / 39
41 3. Injective constructions for S-semigroups (Constructions) Definition: Nucleus A closure operator j on an S-semigroup S A is called a nucleus if it is a subhomomorphism. We denote A j = {a A a = j(a)}. Proposition Let ( S A,, ) be an S-semigroup, j a nucleus on it. Then (A j, j, j ) is again an S-semigroup equipped with the multiplication and action as a j b = j(a b), s j a = j(s a), for any a, b A, s S. In addition, if A is an S-semigroup quantale, then (A j, j, j ) is an S-semigroup quantale as well. Jan Paseka (MU) / 39
42 3. Injective hulls for S-semigroups (Constructions) the nucleus cl: Let ( S A,, ) be an S-semigroup, D S P(A). Define cl(d) := {x A ( a, c A 1, b A, s S) a D c b = a x c b, s D b = s x b}, where A 1 is the monoid obtained from the semigroup A by externally adjoining the identity element 1. Then cl : P(A) P(A) is a nucleus on SP(A) satisfying cl(x ) = x, x A. Jan Paseka (MU) / 39
43 3. Injective hulls for S-semigroups Theorem Theorem 9 (2017 Zhang, Paseka) Let ( S A,, ) be an S-semigroup, cl: S P(A) S P(A) be defined as above. Then (P(A) cl, cl, cl, ) is the ε 0 -injective hull of the S-semigroup S A. Jan Paseka (MU) / 39
44 Conclusion Remarks We have described injectives in the category Ssgr of S-semigroups and showed that every S-semigroup has an ε 0 -injective hull. Based on these results the next step in the future would be to obtain corresponding results in the category of residuated S-semigroups. Jan Paseka (MU) / 39
45 References [1] Adámek J., Herrlich H. and Strecker G. E., Abstract and concrete categories: The joy of cats, John Wiley and Sons, New York, [2] Banaschewski B., Bruns G., Categorical construction of the MacNeille completion, Arch. Math., 1967, [3] Bruns G., Lakser H. Injective hulls of semilattices. Canad. Math. Bull., 1970, 13, [4] Fakhruddin S.M., On the category of S-posets, Acta Sci. Math., 1988, 52, [5] Johnson C.S., J.R., McMorris F.R., Injective hulls on certain S-systems over a semilattice, Proc. Amer. Math. Soc., 1972, 32, [6] Kruml D., Paseka J., Algebraic and categorical aspects of quantales, In: Handbook of Algebra, vol. 5, pp , Elsevier [7] Lambek J., Barr M., Kennison J.F. and Raphael R., Injective hulls of partially ordered monoids, Theory Appl. Categ., 2012, 26, [8] Rosenthal K.I., Quantales and their applications. Pitman Research Notes in Mathematics 234, Harlow, Essex, Jan Paseka (MU) / 39
46 References [9] Rasouli H., Completion of S-posets, Semigroup Forum, 2012, 85, [10] Schein B.M., Injectives in certain classes of semigroups, Semigroup Forum, 1974, 9, [11] Solovyov S., A representation theorem for quantale algebras, Contr. Gen. Alg., 2008, 18, [12] Xia C.C., Zhao B., Han S.W., A Note on injective hulls of posemigropus, Theory Appl. Categ., 2017, 32, [13] Zhang X., Laan V., On injective hulls of S-posets, Semigroup Forum, 2015, 91, [14] Zhang X., Laan V., Injective hulls for posemigroups, Proc. Est. Acad. Sci., 2014, 63, [15] Zhang X., Laan V., Injective hulls for ordered algebras, Algebra Universalis, 2016, 76, Jan Paseka (MU) / 39
47 Thank you! Jan Paseka (MU) / 39
Injective hulls for posemigroups
Proceedings of the stonian Academy of Sciences, 2014, 63, 4, 372 378 doi: 10.3176/proc.2014.4.02 Available online at www.eap.ee/proceedings Injective hulls for posemigroups Xia Zhang a,b and Valdis Laan
More informationFREE OBJECTS OVER POSEMIGROUPS IN THE CATEGORY POSGR
Theory and Applications of Categories, Vol. 32, No. 32, 2017, pp. 1098 1106. FREE OBJECTS OVER POSEMIGROUPS IN THE CATEGORY POSGR SHENGWEI HAN, CHANGCHUN XIA, XIAOJUAN GU ABSTRACT. As we all know, the
More informationA representation theorem for quantale valued sup-algebras
A representation theorem for quantale valued sup-algebras arxiv:1810.09561v1 [math.lo] 22 Oct 2018 Jan Paseka Department of Mathematics and Statistics Faculty of Science, Masaryk University Kotlářská 2,
More informationOn morphisms of lattice-valued formal contexts
On morphisms of lattice-valued formal contexts Sergejs Solovjovs Masaryk University 1/37 On morphisms of lattice-valued formal contexts Sergejs Solovjovs Department of Mathematics and Statistics, Faculty
More informationOn Augmented Posets And (Z 1, Z 1 )-Complete Posets
On Augmented Posets And (Z 1, Z 1 )-Complete Posets Mustafa Demirci Akdeniz University, Faculty of Sciences, Department of Mathematics, 07058-Antalya, Turkey, e-mail: demirci@akdeniz.edu.tr July 11, 2011
More informationON SOME BASIC CONSTRUCTIONS IN CATEGORIES OF QUANTALE-VALUED SUP-LATTICES. 1. Introduction
Math. Appl. 5 (2016, 39 53 DOI: 10.13164/ma.2016.04 ON SOME BASIC CONSTRUCTIONS IN CATEGORIES OF QUANTALE-VALUED SUP-LATTICES RADEK ŠLESINGER Abstract. If the standard concepts of partial-order relation
More informationThe quantale of Galois connections
The quantale of Galois connections Jorge Picado Mathematics Department - University of Coimbra PORTUGAL June 2005 II Conference on Algebraic and Topological Methods in Non-Classical Logics 1 GALOIS CONNECTIONS
More informationOn the normal completion of a Boolean algebra
Journal of Pure and Applied Algebra 181 (2003) 1 14 www.elsevier.com/locate/jpaa On the normal completion of a Boolean algebra B. Banaschewski a, M.M. Ebrahimi b, M. Mahmoudi b; a Department of Mathematics
More informationCategory Theory (UMV/TK/07)
P. J. Šafárik University, Faculty of Science, Košice Project 2005/NP1-051 11230100466 Basic information Extent: 2 hrs lecture/1 hrs seminar per week. Assessment: Written tests during the semester, written
More informationarxiv: v1 [cs.lo] 4 Sep 2018
A characterization of the consistent Hoare powerdomains over dcpos Zhongxi Zhang a,, Qingguo Li b, Nan Zhang a a School of Computer and Control Engineering, Yantai University, Yantai, Shandong, 264005,
More informationCONGRUENCES OF STRONGLY MORITA EQUIVALENT POSEMIGROUPS. 1. Introduction
CONGRUENCES OF STRONGLY MORITA EQUIVALENT POSEMIGROUPS T. TÄRGLA and V. LAAN Abstract. We prove that congruence lattices of strongly Morita equivalent posemigroups with common joint weak local units are
More informationNotes about Filters. Samuel Mimram. December 6, 2012
Notes about Filters Samuel Mimram December 6, 2012 1 Filters and ultrafilters Definition 1. A filter F on a poset (L, ) is a subset of L which is upwardclosed and downward-directed (= is a filter-base):
More informationCONGRUENCES OF STRONGLY MORITA EQUIVALENT POSEMIGROUPS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXXI, 2 (2012), pp. 247 254 247 CONGRUENCES OF STRONGLY MORITA EQUIVALENT POSEMIGROUPS T. TÄRGLA and V. LAAN Abstract. We prove that congruence lattices of strongly Morita
More informationState Transitions as Morphisms for Complete Lattices
State Transitions as Morphisms for Complete attices Bob Coecke FUND-DWIS, Free University of Brussels, Pleinlaan 2, B-1050 Brussels, Belgium. bocoecke@vub.ac.be Isar Stubbe AGE-MAPA, Université Catholique
More informationMV-algebras and fuzzy topologies: Stone duality extended
MV-algebras and fuzzy topologies: Stone duality extended Dipartimento di Matematica Università di Salerno, Italy Algebra and Coalgebra meet Proof Theory Universität Bern April 27 29, 2011 Outline 1 MV-algebras
More informationDominions, zigzags and epimorphisms for partially ordered semigroups
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 18, Number 1, June 2014 Available online at http://acutmmathutee Dominions, zigzags and epimorphisms for partially ordered semigroups
More informationCartesian Closed Topological Categories and Tensor Products
Cartesian Closed Topological Categories and Tensor Products Gavin J. Seal October 21, 2003 Abstract The projective tensor product in a category of topological R-modules (where R is a topological ring)
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato September 10, 2015 ABSTRACT. This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a
More informationSPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM
SPECTRAL-LIKE DUALITY FOR DISTRIBUTIVE HILBERT ALGEBRAS WITH INFIMUM SERGIO A. CELANI AND MARÍA ESTEBAN Abstract. Distributive Hilbert Algebras with infimum, or DH -algebras, are algebras with implication
More informationRelational semantics for a fragment of linear logic
Relational semantics for a fragment of linear logic Dion Coumans March 4, 2011 Abstract Relational semantics, given by Kripke frames, play an essential role in the study of modal and intuitionistic logic.
More informationSymbol Index Group GermAnal Ring AbMonoid
Symbol Index 409 Symbol Index Symbols of standard and uncontroversial usage are generally not included here. As in the word index, boldface page-numbers indicate pages where definitions are given. If a
More informationCONTINUITY. 1. Continuity 1.1. Preserving limits and colimits. Suppose that F : J C and R: C D are functors. Consider the limit diagrams.
CONTINUITY Abstract. Continuity, tensor products, complete lattices, the Tarski Fixed Point Theorem, existence of adjoints, Freyd s Adjoint Functor Theorem 1. Continuity 1.1. Preserving limits and colimits.
More informationON THE CONGRUENCE LATTICE OF A FRAME
PACIFIC JOURNAL OF MATHEMATICS Vol. 130, No. 2,1987 ON THE CONGRUENCE LATTICE OF A FRAME B. BANASCHEWSKI, J. L. FRITH AND C. R. A. GILMOUR Recall that the Skula modification SkX of a topological space
More informationACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes)
ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes) Steve Vickers CS Theory Group Birmingham 2. Theories and models Categorical approach to many-sorted
More informationMorita equivalence of many-sorted algebraic theories
Journal of Algebra 297 (2006) 361 371 www.elsevier.com/locate/jalgebra Morita equivalence of many-sorted algebraic theories Jiří Adámek a,,1, Manuela Sobral b,2, Lurdes Sousa c,3 a Department of Theoretical
More informationLecture 1: Lattice(I)
Discrete Mathematics (II) Spring 207 Lecture : Lattice(I) Lecturer: Yi Li Lattice is a special algebra structure. It is also a part of theoretic foundation of model theory, which formalizes the semantics
More informationMORITA EQUIVALENCE OF MANY-SORTED ALGEBRAIC THEORIES
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 04 39 MORITA EQUIVALENCE OF MANY-SORTED ALGEBRAIC THEORIES JIŘÍ ADÁMEK, MANUELA SOBRAL AND LURDES SOUSA Abstract: Algebraic
More informationOn complete objects in the category of T 0 closure spaces
@ Applied General Topology c Universidad Politécnica de Valencia Volume 4, No. 1, 2003 pp. 25 34 On complete objects in the category of T 0 closure spaces D. Deses, E. Giuli and E. Lowen-Colebunders Abstract.
More informationCOMPACT ORTHOALGEBRAS
COMPACT ORTHOALGEBRAS ALEXANDER WILCE Abstract. We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and
More informationClosure operators on sets and algebraic lattices
Closure operators on sets and algebraic lattices Sergiu Rudeanu University of Bucharest Romania Closure operators are abundant in mathematics; here are a few examples. Given an algebraic structure, such
More informationINJECTIVE HULLS OF PARTIALLY ORDERED MONOIDS
Theory and Applications of Categories, Vol. 26, No. 13, 2012, pp. 338 348. INJECTIVE HULLS OF PARTIALLY ORDERED MONOIDS J. LAMBEK MICHAEL BARR, JOHN F. KENNISON, R. RAPHAEL Abstract. We find the injective
More informationarxiv: v1 [math.ct] 28 Oct 2017
BARELY LOCALLY PRESENTABLE CATEGORIES arxiv:1710.10476v1 [math.ct] 28 Oct 2017 L. POSITSELSKI AND J. ROSICKÝ Abstract. We introduce a new class of categories generalizing locally presentable ones. The
More informationCONSTRUCTIVE GELFAND DUALITY FOR C*-ALGEBRAS
CONSTRUCTIVE GELFAND DUALITY FOR C*-ALGEBRAS THIERRY COQUAND COMPUTING SCIENCE DEPARTMENT AT GÖTEBORG UNIVERSITY AND BAS SPITTERS DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, EINDHOVEN UNIVERSITY OF
More informationLattice Theory Lecture 5. Completions
Lattice Theory Lecture 5 Completions John Harding New Mexico State University www.math.nmsu.edu/ JohnHarding.html jharding@nmsu.edu Toulouse, July 2017 Completions Definition A completion of a poset P
More informationBoolean Algebra and Propositional Logic
Boolean Algebra and Propositional Logic Takahiro Kato June 23, 2015 This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection
More informationCoreflections in Algebraic Quantum Logic
Coreflections in Algebraic Quantum Logic Bart Jacobs Jorik Mandemaker Radboud University, Nijmegen, The Netherlands Abstract Various generalizations of Boolean algebras are being studied in algebraic quantum
More informationThe role of the overlap relation in constructive mathematics
The role of the overlap relation in constructive mathematics Francesco Ciraulo Department of Mathematics and Computer Science University of PALERMO (Italy) ciraulo@math.unipa.it www.math.unipa.it/ ciraulo
More informationUniversal Algebra for Logics
Universal Algebra for Logics Joanna GRYGIEL University of Czestochowa Poland j.grygiel@ajd.czest.pl 2005 These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic
More informationUsing topological systems to create a framework for institutions
Using topological systems to create a framework for institutions Sergejs Solovjovs Brno University of Technology 1/34 Using topological systems to create a framework for institutions Jeffrey T. Denniston
More informationSpan, Cospan, and Other Double Categories
ariv:1201.3789v1 [math.ct] 18 Jan 2012 Span, Cospan, and Other Double Categories Susan Nieield July 19, 2018 Abstract Given a double category D such that D 0 has pushouts, we characterize oplax/lax adjunctions
More informationCATEGORY THEORY. Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths.
CATEGORY THEORY PROFESSOR PETER JOHNSTONE Cats have been around for 70 years. Eilenberg + Mac Lane =. Cats are about building bridges between different parts of maths. Definition 1.1. A category C consists
More informationSkew Boolean algebras
Skew Boolean algebras Ganna Kudryavtseva University of Ljubljana Faculty of Civil and Geodetic Engineering IMFM, Ljubljana IJS, Ljubljana New directions in inverse semigroups Ottawa, June 2016 Plan of
More informationCategorical lattice-valued topology Lecture 1: powerset and topological theories, and their models
Categorical lattice-valued topology Lecture 1: powerset and topological theories, and their models Sergejs Solovjovs Department of Mathematics and Statistics, Faculty of Science, Masaryk University Kotlarska
More informationCHARACTERIZATIONS OF L-CONVEX SPACES
Iranian Journal of Fuzzy Systems Vol 13, No 4, (2016) pp 51-61 51 CHARACTERIZATIONS OF L-CONVEX SPACES B PANG AND Y ZHAO Abstract In this paper, the concepts of L-concave structures, concave L- interior
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationAlgebraic Properties of the Category of Q-P Quantale Modules. LIANG Shaohui [a],*
Progress in Applied Mathematics Vol. 6, No. 1, 2013, pp. [54 63] DOI: 10.3968/j.pam.1925252820130601.409 ISSN 1925-251X [Print] ISSN 1925-2528 [Online] www.cscanada.net www.cscanada.org Algebraic Properties
More informationOn Morita equivalence of partially ordered semigroups with local units
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 15, Number 2, 2011 Available online at www.math.ut.ee/acta/ On Morita equivalence of partially ordered semigroups with local units
More informationHouston Journal of Mathematics. c 2004 University of Houston Volume 30, No. 4, 2004
Houston Journal of Mathematics c 2004 University of Houston Volume 30, No. 4, 2004 MACNEILLE COMPLETIONS OF HEYTING ALGEBRAS JOHN HARDING AND GURAM BEZHANISHVILI Communicated by Klaus Kaiser Abstract.
More informationAn embedding of ChuCors in L-ChuCors
Proceedings of the 10th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2010 27 30 June 2010. An embedding of ChuCors in L-ChuCors Ondrej Krídlo 1,
More informationRESIDUATION SUBREDUCTS OF POCRIGS
Bulletin of the Section of Logic Volume 39:1/2 (2010), pp. 11 16 Jānis Cīrulis RESIDUATION SUBREDUCTS OF POCRIGS Abstract A pocrig (A,,, 1) is a partially ordered commutative residuated integral groupoid.
More informationDUAL BCK-ALGEBRA AND MV-ALGEBRA. Kyung Ho Kim and Yong Ho Yon. Received March 23, 2007
Scientiae Mathematicae Japonicae Online, e-2007, 393 399 393 DUAL BCK-ALGEBRA AND MV-ALGEBRA Kyung Ho Kim and Yong Ho Yon Received March 23, 2007 Abstract. The aim of this paper is to study the properties
More informationBases as Coalgebras. Bart Jacobs
Bases as Coalgebras Bart Jacobs Institute for Computing and Information Sciences (icis), Radboud University Nijmegen, The Netherlands. Webaddress: www.cs.ru.nl/b.jacobs Abstract. The free algebra adjunction,
More informationarxiv: v1 [math.lo] 30 Aug 2018
arxiv:1808.10324v1 [math.lo] 30 Aug 2018 Real coextensions as a tool for constructing triangular norms Thomas Vetterlein Department of Knowledge-Based Mathematical Systems Johannes Kepler University Linz
More informationA note on separation and compactness in categories of convergence spaces
@ Applied General Topology c Universidad Politécnica de Valencia Volume 4, No. 1, 003 pp. 1 13 A note on separation and compactness in categories of convergence spaces Mehmet Baran and Muammer Kula Abstract.
More informationCross Connection of Boolean Lattice
International Journal of Algebra, Vol. 11, 2017, no. 4, 171-179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7419 Cross Connection of Boolean Lattice P. G. Romeo P. R. Sreejamol Dept.
More informationA Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries
A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries Johannes Marti and Riccardo Pinosio Draft from April 5, 2018 Abstract In this paper we present a duality between nonmonotonic
More information2. ETALE GROUPOIDS MARK V. LAWSON
2. ETALE GROUPOIDS MARK V. LAWSON Abstract. In this article, we define étale groupoids and describe some of their properties. 1. Generalities 1.1. Categories. A category is usually regarded as a category
More informationAlgebra and local presentability: how algebraic are they? (A survey)
Algebra and local presentability: how algebraic are they? (A survey) Jiří Adámek 1 and Jiří Rosický 2, 1 Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague,
More informationON THE UNIFORMIZATION OF L-VALUED FRAMES
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 09 42 ON THE UNIFORMIZATION OF L-VALUED FRAMES J. GUTIÉRREZ GARCÍA, I. MARDONES-PÉREZ, JORGE PICADO AND M. A. DE PRADA
More informationPART I. Abstract algebraic categories
PART I Abstract algebraic categories It should be observed first that the whole concept of category is essentially an auxiliary one; our basic concepts are those of a functor and a natural transformation.
More informationVarieties of Heyting algebras and superintuitionistic logics
Varieties of Heyting algebras and superintuitionistic logics Nick Bezhanishvili Institute for Logic, Language and Computation University of Amsterdam http://www.phil.uu.nl/~bezhanishvili email: N.Bezhanishvili@uva.nl
More informationA duality-theoretic approach to MTL-algebras
A duality-theoretic approach to MTL-algebras Sara Ugolini (Joint work with W. Fussner) BLAST 2018 - Denver, August 6th 2018 A commutative, integral residuated lattice, or CIRL, is a structure A = (A,,,,,
More informationOn regularity of sup-preserving maps: generalizing Zareckiĭ s theorem
Semigroup Forum (2011) 83:313 319 DOI 10.1007/s00233-011-9311-0 RESEARCH ARTICLE On regularity o sup-preserving maps: generalizing Zareckiĭ s theorem Ulrich Höhle Tomasz Kubiak Received: 7 March 2011 /
More informationA Natural Equivalence for the Category of Coherent Frames
A Natural Equivalence for the Category of Coherent Frames Wolf Iberkleid and Warren Wm. McGovern Abstract. The functor on the category of bounded lattices induced by reversing their order, gives rise to
More informationCLOSURE, INTERIOR AND NEIGHBOURHOOD IN A CATEGORY
CLOSURE, INTERIOR AND NEIGHBOURHOOD IN A CATEGORY DAVID HOLGATE AND JOSEF ŠLAPAL Abstract. While there is extensive literature on closure operators in categories, less has been done in terms of their dual
More informationCommutative Algebra Lecture 3: Lattices and Categories (Sept. 13, 2013)
Commutative Algebra Lecture 3: Lattices and Categories (Sept. 13, 2013) Navid Alaei September 17, 2013 1 Lattice Basics There are, in general, two equivalent approaches to defining a lattice; one is rather
More informationNotes on Ordered Sets
Notes on Ordered Sets Mariusz Wodzicki September 10, 2013 1 Vocabulary 1.1 Definitions Definition 1.1 A binary relation on a set S is said to be a partial order if it is reflexive, x x, weakly antisymmetric,
More informationCongruence Boolean Lifting Property
Congruence Boolean Lifting Property George GEORGESCU and Claudia MUREŞAN University of Bucharest Faculty of Mathematics and Computer Science Academiei 14, RO 010014, Bucharest, Romania Emails: georgescu.capreni@yahoo.com;
More informationUniversity of Oxford, Michaelis November 16, Categorical Semantics and Topos Theory Homotopy type theor
Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011 November 16, 2011 References Johnstone, P.T.: Sketches of an Elephant. A Topos-Theory Compendium.
More informationA bitopological point-free approach to compactifications
A bitopological point-free approach to compactifications Olaf Karl Klinke a, Achim Jung a, M. Andrew Moshier b a School of Computer Science University of Birmingham Birmingham, B15 2TT England b School
More informationBoolean Algebras, Boolean Rings and Stone s Representation Theorem
Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to
More informationWhat are Iteration Theories?
What are Iteration Theories? Jiří Adámek and Stefan Milius Institute of Theoretical Computer Science Technical University of Braunschweig Germany adamek,milius @iti.cs.tu-bs.de Jiří Velebil Department
More informationSilvio Valentini Dip. di Matematica - Università di Padova
REPRESENTATION THEOREMS FOR QUANTALES Silvio Valentini Dip. di Matematica - Università di Padova Abstract. In this paper we prove that any quantale Q is (isomorphic to) a quantale of suitable relations
More informationMODELS OF HORN THEORIES
MODELS OF HORN THEORIES MICHAEL BARR Abstract. This paper explores the connection between categories of models of Horn theories and models of finite limit theories. The first is a proper subclass of the
More informationFUZZY ACTS OVER FUZZY SEMIGROUPS AND SHEAVES
Iranian Journal of Fuzzy Systems Vol. 11, No. 4, (2014) pp. 61-73 61 FUZZY ACTS OVER FUZZY SEMIGROUPS AND SHEAVES M. HADDADI Abstract. Although fuzzy set theory and sheaf theory have been developed and
More informationAn algebraic approach to Gelfand Duality
An algebraic approach to Gelfand Duality Guram Bezhanishvili New Mexico State University Joint work with Patrick J Morandi and Bruce Olberding Stone = zero-dimensional compact Hausdorff spaces and continuous
More information1 Differentiable manifolds and smooth maps
1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set
More informationMetric spaces and textures
@ Appl. Gen. Topol. 18, no. 1 (2017), 203-217 doi:10.4995/agt.2017.6889 c AGT, UPV, 2017 Metric spaces and textures Şenol Dost Hacettepe University, Department of Mathematics and Science Education, 06800
More informationConstructive version of Boolean algebra
Constructive version of Boolean algebra Francesco Ciraulo, Maria Emilia Maietti, Paola Toto Abstract The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean
More informationANNIHILATOR IDEALS IN ALMOST SEMILATTICE
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 7(2017), 339-352 DOI: 10.7251/BIMVI1702339R Former BULLETIN
More informationTopology Proceedings. COPYRIGHT c by Topology Proceedings. All rights reserved.
Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: topolog@auburn.edu ISSN: 0146-4124
More informationSIMPLE LOGICS FOR BASIC ALGEBRAS
Bulletin of the Section of Logic Volume 44:3/4 (2015), pp. 95 110 http://dx.doi.org/10.18778/0138-0680.44.3.4.01 Jānis Cīrulis SIMPLE LOGICS FOR BASIC ALGEBRAS Abstract An MV-algebra is an algebra (A,,,
More informationFuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras
Fuzzy filters and fuzzy prime filters of bounded Rl-monoids and pseudo BL-algebras Jiří Rachůnek 1 Dana Šalounová2 1 Department of Algebra and Geometry, Faculty of Sciences, Palacký University, Tomkova
More informationLogical connections in the many-sorted setting
Logical connections in the many-sorted setting Jiří Velebil Czech Technical University in Prague Czech Republic joint work with Alexander Kurz University of Leicester United Kingdom AK & JV AsubL4 1/24
More informationContinuity of partially ordered soft sets via soft Scott topology and soft sobrification A. F. Sayed
Bulletin of Mathematical Sciences and Applications Online: 2014-08-04 ISSN: 2278-9634, Vol. 9, pp 79-88 doi:10.18052/www.scipress.com/bmsa.9.79 2014 SciPress Ltd., Switzerland Continuity of partially ordered
More informationA topological duality for posets
A topological duality for posets RAMON JANSANA Universitat de Barcelona join work with Luciano González TACL 2015 Ischia, June 26, 2015. R. Jansana 1 / 20 Introduction In 2014 M. A. Moshier and P. Jipsen
More information(IC)LM-FUZZY TOPOLOGICAL SPACES. 1. Introduction
Iranian Journal of Fuzzy Systems Vol. 9, No. 6, (2012) pp. 123-133 123 (IC)LM-FUZZY TOPOLOGICAL SPACES H. Y. LI Abstract. The aim of the present paper is to define and study (IC)LM-fuzzy topological spaces,
More informationThe category of linear modular lattices
Bull. Math. Soc. Sci. Math. Roumanie Tome 56(104) No. 1, 2013, 33 46 The category of linear modular lattices by Toma Albu and Mihai Iosif Dedicated to the memory of Nicolae Popescu (1937-2010) on the occasion
More informationKleene algebras with implication
Kleene algebras with implication Hernán Javier San Martín CONICET Departamento de Matemática, Facultad de Ciencias Exactas, UNLP September 2016 Hernán Javier San Martín (UNLP) PC September 2016 1 / 16
More informationEmbedding theorems for normal divisible residuated lattices
Embedding theorems for normal divisible residuated lattices Chapman University Department of Mathematics and Computer Science Orange, California University of Siena Department of Mathematics and Computer
More informationQuotient Structure of Interior-closure Texture Spaces
Filomat 29:5 (2015), 947 962 DOI 10.2298/FIL1505947D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Quotient Structure of Interior-closure
More informationFibres. Temesghen Kahsai. Fibres in Concrete category. Generalized Fibres. Fibres. Temesghen Kahsai 14/02/ 2007
14/02/ 2007 Table of Contents ... and back to theory Example Let Σ = (S, TF) be a signature and Φ be a set of FOL formulae: 1. SPres is the of strict presentation with: objects: < Σ, Φ >, morphism σ :
More informationOn the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P
Order 18: 49 60, 2001. 2001 Kluwer Academic Publishers. Printed in the Netherlands. 49 On the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P ROZÁLIA SZ. MADARÁSZ Institute of Mathematics,
More informationlanguages by semifilter-congruences
ideas Suffix semifilter-congruences Southwest Univ. Southwest Univ. Hongkong Univ. July 5 9, 2010, Nankai, China. Prefixsuffix Contents ideas 1 2 ideas 3 Suffix- 4 Prefix-suffix- Suffix Prefixsuffix ideas
More informationCategorical relativistic quantum theory. Chris Heunen Pau Enrique Moliner Sean Tull
Categorical relativistic quantum theory Chris Heunen Pau Enrique Moliner Sean Tull 1 / 15 Idea Hilbert modules: naive quantum field theory Idempotent subunits: base space in any category Support: where
More informationREPRESENTATION THEOREMS FOR IMPLICATION STRUCTURES
Wojciech Buszkowski REPRESENTATION THEOREMS FOR IMPLICATION STRUCTURES Professor Rasiowa [HR49] considers implication algebras (A,, V ) such that is a binary operation on the universe A and V A. In particular,
More informationDuality and Automata Theory
Duality and Automata Theory Mai Gehrke Université Paris VII and CNRS Joint work with Serge Grigorieff and Jean-Éric Pin Elements of automata theory A finite automaton a 1 2 b b a 3 a, b The states are
More informationSTRATIFIED (L, M)-FUZZY Q-CONVERGENCE SPACES
Iranian Journal of Fuzzy Systems Vol. 13, No. 4, (2016) pp. 95-111 95 STRATIFIED (L, M)-FUZZY Q-CONVERGENCE SPACES B. PANG AND Y. ZHAO Abstract. This paper presents the concepts of (L, M)-fuzzy Q-convergence
More informationMTH 428/528. Introduction to Topology II. Elements of Algebraic Topology. Bernard Badzioch
MTH 428/528 Introduction to Topology II Elements of Algebraic Topology Bernard Badzioch 2016.12.12 Contents 1. Some Motivation.......................................................... 3 2. Categories
More informationWhen does a semiring become a residuated lattice?
When does a semiring become a residuated lattice? Ivan Chajda and Helmut Länger arxiv:1809.07646v1 [math.ra] 20 Sep 2018 Abstract It is an easy observation that every residuated lattice is in fact a semiring
More information