Representing integral quantales and residuated lattices by tolerances by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with Kalle Kaarli, Tartu Univ.) SSAOS 54, Trojanovice, Czech Republic, Sept. 3-9, 2016. by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) Quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multplicative lattices of ideals from ring theory. Residuated lattices were introduced by Dilworth and Ward and they are used in several branches of mathematics, including areas of ideal lattices of rings, lattice-ordered groups, multivalued logic and formal languages. In our lecture, we try to show that integral quantales and complete integral residuated lattices are strongly related with the complete tolerances of their underlying lattice. 1. Background and preliminaries by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) In our lecture, we try to show that integral quantales and complete integral residuated lattices are strongly related with the complete tolerances of their underlying lattice. 1. Background and preliminaries Quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multplicative lattices of ideals from ring theory. Residuated lattices were introduced by Dilworth and Ward and they are used in several branches of mathematics, including areas of ideal lattices of rings, lattice-ordered groups, multivalued logic and formal languages. by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) 1. Background and preliminaries Quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multplicative lattices of ideals from ring theory. Residuated lattices were introduced by Dilworth and Ward and they are used in several branches of mathematics, including areas of ideal lattices of rings, lattice-ordered groups, multivalued logic and formal languages. In our lecture, we try to show that integral quantales and complete integral residuated lattices are strongly related with the complete tolerances of their underlying lattice. by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) Definition 1. A quantale is an algebraic structure Q = (L; _; ), such that (L; ≤) is a complete lattice (induced by the join operation _) and (L; ) is a semigroup satisfying W W W W a bi = (a bi ) and bi a = (bi a). i2I i2I i2I i2I for all a 2 L and bi 2 L, i 2 I . Q is called commutative, if is commutative, and Q is unital, whenever (L; ) is a monoid. A unital quantale in which the neutral element of coincides to the greatest element 1 of the lattice L is called integral. Hence in any integral quantale 1 x = x 1 = x (1.1) A subset K ⊆ L is called a subquantale of Q if it is closed under arbitrary joins and . General notions by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) A quantale is an algebraic structure Q = (L; _; ), such that (L; ≤) is a complete lattice (induced by the join operation _) and (L; ) is a semigroup satisfying W W W W a bi = (a bi ) and bi a = (bi a). i2I i2I i2I i2I for all a 2 L and bi 2 L, i 2 I . Q is called commutative, if is commutative, and Q is unital, whenever (L; ) is a monoid. A unital quantale in which the neutral element of coincides to the greatest element 1 of the lattice L is called integral. Hence in any integral quantale 1 x = x 1 = x (1.1) A subset K ⊆ L is called a subquantale of Q if it is closed under arbitrary joins and . General notions Definition 1. by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) Q is called commutative, if is commutative, and Q is unital, whenever (L; ) is a monoid. A unital quantale in which the neutral element of coincides to the greatest element 1 of the lattice L is called integral. Hence in any integral quantale 1 x = x 1 = x (1.1) A subset K ⊆ L is called a subquantale of Q if it is closed under arbitrary joins and . General notions Definition 1. A quantale is an algebraic structure Q = (L; _; ), such that (L; ≤) is a complete lattice (induced by the join operation _) and (L; ) is a semigroup satisfying W W W W a bi = (a bi ) and bi a = (bi a). i2I i2I i2I i2I for all a 2 L and bi 2 L, i 2 I . by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) A subset K ⊆ L is called a subquantale of Q if it is closed under arbitrary joins and . General notions Definition 1. A quantale is an algebraic structure Q = (L; _; ), such that (L; ≤) is a complete lattice (induced by the join operation _) and (L; ) is a semigroup satisfying W W W W a bi = (a bi ) and bi a = (bi a). i2I i2I i2I i2I for all a 2 L and bi 2 L, i 2 I . Q is called commutative, if is commutative, and Q is unital, whenever (L; ) is a monoid. A unital quantale in which the neutral element of coincides to the greatest element 1 of the lattice L is called integral. Hence in any integral quantale 1 x = x 1 = x (1.1) by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) General notions Definition 1. A quantale is an algebraic structure Q = (L; _; ), such that (L; ≤) is a complete lattice (induced by the join operation _) and (L; ) is a semigroup satisfying W W W W a bi = (a bi ) and bi a = (bi a). i2I i2I i2I i2I for all a 2 L and bi 2 L, i 2 I . Q is called commutative, if is commutative, and Q is unital, whenever (L; ) is a monoid. A unital quantale in which the neutral element of coincides to the greatest element 1 of the lattice L is called integral. Hence in any integral quantale 1 x = x 1 = x (1.1) A subset K ⊆ L is called a subquantale of Q if it is closed under arbitrary joins and . by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) (b) The two-sided ideals of a ring (R; +; ·) with unit form an integral quantale (I(R); _; •), where (I(R); _; \) is the complete lattice of the ideals of R and • is their usual multiplication, i.e. for any I ; J 2 I(R) we P have I • J := f fini · j j i 2 I , j 2 Jg. Definition 2. A residuated lattice is an algebra L = (L; _; ^; ; n; =; 1) of type (2,2,2,2,2,0) such that (i)( L; _; ^) is a lattice, (ii)( L; ) is a semigroup satisfying 1 x = x 1 = x, for all x 2 L. (iii) L satisfies the adjointness properties, that is, for all x; y; z 2 L x y ≤ z , y ≤ xnz , x ≤ z=y: L is called commutative, if is commutative. In this case xny and y=x being equal, they are denoted as x ! y. Examples 1. (a) Frames are commutative quantales in which and the meet operation ^ coincide. by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) Definition 2. A residuated lattice is an algebra L = (L; _; ^; ; n; =; 1) of type (2,2,2,2,2,0) such that (i)( L; _; ^) is a lattice, (ii)( L; ) is a semigroup satisfying 1 x = x 1 = x, for all x 2 L. (iii) L satisfies the adjointness properties, that is, for all x; y; z 2 L x y ≤ z , y ≤ xnz , x ≤ z=y: L is called commutative, if is commutative. In this case xny and y=x being equal, they are denoted as x ! y. Examples 1. (a) Frames are commutative quantales in which and the meet operation ^ coincide. (b) The two-sided ideals of a ring (R; +; ·) with unit form an integral quantale (I(R); _; •), where (I(R); _; \) is the complete lattice of the ideals of R and • is their usual multiplication, i.e. for any I ; J 2 I(R) we P have I • J := f fini · j j i 2 I , j 2 Jg. by S´andor Radeleczki, Math. Institute, Univ. of Miskolc (joint research with KalleAggregation Kaarli, Tartu Univ.) L is called commutative, if is commutative. In this case xny and y=x being equal, they are denoted as x ! y. Examples 1. (a) Frames are commutative quantales in which and the meet operation ^ coincide. (b) The two-sided ideals of a ring (R; +; ·) with unit form an integral quantale (I(R); _; •), where (I(R); _; \) is the complete lattice of the ideals of R and • is their usual multiplication, i.e. for any I ; J 2 I(R) we P have I • J := f fini · j j i 2 I , j 2 Jg. Definition 2. A residuated lattice is an algebra L = (L; _; ^; ; n; =; 1) of type (2,2,2,2,2,0) such that (i)( L; _; ^) is a lattice, (ii)( L; ) is a semigroup satisfying 1 x = x 1 = x, for all x 2 L. (iii) L satisfies the adjointness properties, that is, for all x; y; z 2 L x y ≤ z , y ≤ xnz , x ≤ z=y: by S´andor Radeleczki, Math.