 Okay, in this talk I will present a joint work research project that I start with Olivia Caramello and the aim of this research project is to study the context and the theory of MV algebras from a topist theoretic point of view. And what does it we obtain until now? We start to study categorical equivalents that I already know in the theory in the context of MV algebras and what we do is what's already mentioned Olivia in this course, we start with this categorical equivalence and another question is is this Morita equivalence induced and Morita equivalence between the theories that are involved. And we ask for in affirmative way for these two categorical equivalents that we study. We study in particular Mundich's equivalence which is a categorical equivalence between the category of MV algebras and the category of Lattice ordinal million groups with strong units that you know the name of the structure is very long so I will come for short just L groups with strong unit. And the other categorical equivalence that we look at is the knowledge equivalence that proved that the equivalence between the category of perfect MV algebras and the category of L groups and what we do once we prove that this we can actually lift these two categorical equivalences is to of course apply the bridge technique to these equivalences because as Olivia said it is very nice once you have these and Morita equivalences you can apply the bridge technique in order to transfer information from one theory to the other to the other so just obtain new information on the theories. And in recent work we actually work in the opposite way we actually construct new classes of Morita equivalences that in among among this class we obtain also as a particular the the knowledge Lattice equivalence. So one of these Morita equivalences that we construct is indeed the knowledge is the lifting of the knowledge equivalence but the results that we obtain in connection to the lifting of Munch's equivalence applying our bridge technique is the first one the first immediate result is that the theory of L groups with strong unit is a theory of pre-shift type and we obtain this result very easily because this is this theory is Morita equivalence to an algebraic theory that is the theory of MV algebras and of course we are considered other invariants and we obtain also the second result that say that the questions of these two theories are in rejection. This let me remark the fact that this result cannot be obtained just consider the equivalence at the semantic level because what does the Munch's equivalence say? It says just that if we restrict the categorical domain let me say we have a restriction on the categorical codomain and we have that these two categories are still equivalent but here we have more. We say that if we had for example some action to the theory of MV algebras you know that there are some action that you can add to the theory of L groups with strong units and you obtain two questions of these two theories that are still Morita equivalence and the most important factor and the reality of this result is that these two theories are not by interpretable. Of course if you have a way to translate from the language on theories to the other is these results would be very trivial but in this case we prove that these two theories are not by interpretable and we can obtain in an indirect way this bijection because every as Olivia is saying this in our course every sequence can be written in this form where these formulas are irreducible and since we prove that indeed there is yes it's not exactly in this way but for easier I just write yes here I have to write like as is yes for so this since you have we've proved that there is this characterization for the financially presentable L groups and we prove that indeed that every financially presentable L group to the strong unit is indeed financially presented we have a way to you have a we have a bijection between the reducible formulas that can imply in an indirect way these bijection between the quotients and also we prove a form of compactness and computer is resolved for the theory of L groups with strong units just considering an invariance just transferring some information that we know for the theory of MV algebra so just consider certain invariance and these results are very surprising because the theory of L groups with strong unit is an infinity theory it means that we need of an infinity this junction to express the property to be a strong unit and so you can imagine that prove like that this that an infinity theory is complete at priori is can be very difficult but we obtain this result as just as an application of the bridge of the bridge technique so just transferring some information what we will see the precision when you say completeness it means that it satisfies a completeness feeling with respect to the set based models yes sure sure yes sure sure yes we you say in your course that every geometry in this is complete with respect to models in any universe so here we say completion with respect to set that is not this is the thing that is not true for every geometric theory of course and we prove also a shift erratic version of this Moondish's equivalency indeed we prove that if you consider the the category of models of these two theories in any purpose of shifts on a certain topological space then you have again a categorical equivalence that is natural with respect to the topological space but just let me list the results that we obtain applying the the bridge techniques to the other equivalence that is the lifting of the denominator equivalence we obtain again as an immediate result that the theory of perfect MV algebra is of pre-shift type and this is true because it is more equivalent to an algebraic theory that in this case the theory of L groups and even in this case we prove that these two theories are not by interpretable but if we restrict the they are not globally by interpretable but we found three partial level of by interpretation by considering as many invariance so just considering the invariance we obtain of interpretation at the level of irreducible formulas at the levels of geometric sentences and at the level of imaginary but we will see later how invariance will lead to these results and we obtain also that every financially presentable model of the theory of perfect MV algebra can be is a financially presentable also in the variety that is generated by the perfect MV algebras and of course the complex direction is true because every financially presentable object in the variety is financially presented with respect to the theory of perfect MV algebras but the other science is not trivial and we prove also these results and finally we obtain this representation result for financially generated algebra in the variety generated by perfect MV algebra by studying it's very it's this is the very new technique that we used by studying the the classifying choppers of the theory of perfect MV algebra is more particular by studying the classifying the grounded topology associated with the theory of perfect MV algebra regarded as a quotient of the theory of the variety generated by this perfect MV algebra that I will say in the following I will call just a chance variety for sure but we say that we construct new marite equivalences and we actually say in the opposite way because in this case we want we prove first that the theory is involved with the harrow preshift type and we study the theory of local MV algebras in an arbitrary variety and we prove that globally this theory the is so if we intersect if we consider the theory of local MV algebra in the whole variety of MV algebras this theory is not a preshift type but if we restrict this and we regarded the theory of local MV algebras in any proper sub variety of the variety of MV algebras then these theories are of preshift types and we once and we once we prove that these theories are marite equivalence to certain theories that extend the two rebel groups and see how we prove that we prove because we obtain that these two theories are both of preshift type in this case to obtain the marite equivalence is just is a sufficient to to prove the categorical equivalence of the models of the categories of models at the level of sets but this is just a particular case if both theories are of preshift types and again we obtain that finitely presentable models of these theory of local MV algebra in a proper sub variety are also finitely presentable with as a object in the variety that we choose and again we obtain this representation result for finitely generated algebras in the variety as finite products of local MV algebras of course in the variety but before going into the details of these results let me just recall you the the definition of the structure that are involved in these equivalences both barrage or categorical and so we have the notion of MV algebras that can be seen as that is indeed the semantical counterpart of infinite lucrative logic and so I would like to say that infinite lucrative logic is like given more appropriate description of the reality with respect to classical logic because in reality you have that everything is not just black or white so it's just not true or false but you have a scale of gray that is described very good by this by this logic because you can see for example that one model of this the standard models of this theory is the unit interval zero one where the operation the plus and the negation are defined in this very easy way and the following just for the notation indicate with MV the theory of MV algebras what does it mean by the way this acronym MV many valued yes many values and yes many well the logic you have many value the algebra is very not very uh how you say yes yes uh and creative like I want to say and we have in the particular class of perfect MV of MV algebras is the class of perfect MV algebras that are those algebras that are generated by the radical where for radical you take the standard definition in as intersection of maximal ideals and we indicate with this symbol the theory of perfect MV algebras and we prove that in the particular case for of the variety generated by perfect MV algebras that I call chunks variety we can describe the radical in this very easy way we just buy an equation and this is I want to remark to the on this result because we we see that we obtain just a generalization I can say with this result later and on the other side of course we have the the class of L groups what what is an L group is a group with a partial order structure that is compatible with the group operation that means that you have translation invariant property and an element of this group is called a strong unit if it is positive and it is if it is an infinity if it is an Archimedean element and you can see here in this this freight there is a natural number where the infinity disjunction appear and in the following I will just indicate with L the cheer of L groups and L U the cheer of L groups with strong units so what we obtain we say that for the the first two more each equivalence we have the same general patterns where you can say so we regard the two categorical equivalence equivalences as equivalence of categories of models in the universe of sets and so the further the natural question is this equivalence can be can be regarded can be generalized if we look at the models not just in the universal set but in any in any universe so in any growth endic topos and actually we prove that we can generalize these two factors and still we have a categorical equivalence and this categorical equivalence is natural with respect to the topos because we prove that the construction of these two factors allows also only financial conjunction and it involves only finite finite finite limit and up to equal limit so just geometric construction and of course this is just the definition of Morita equivalence so we have that the theory of the of MV algebra is Morita equivalence to L U and P is Morita equivalence to L but now we finally have these Morita equivalences so now we can apply our bridge technique and so you can see here how we obtained the bijection between quotient we here we use this is just an application if you want of duality theorem of olivia because you duality theorem says that you have a bijection between subtopuses of the classifying topos and quotient of the theory so if you since you have that these two theories are this have the same classifying topos and you regarded as invariance the concept to be a subtopus you have that the quotient of the theory of MV algebra are in bijection with the quotient of the of the theory of value groups as I remarked this as I already remarked this result is not trivial because these two theories are not by interpretable and I explained already before how you can obtain these results in an indirect way so we'll pass to the next result that we obtained by applying bridge technique this is an application of the bridge techniques the second Morita equivalence that we obtained and it's about the partial three partial leave it up by interpretation that we that we approved and you can see that if you just if you consider these three invariance you obtained the bijection the by interpretation at the level of the of the two theories so for example if you consider as an invariance to be an irreducible object the reducible object in the the level of the classifying topos you have here the irreducible formula for the theory p and here the irreducible formulas for the theory l and you obtained the bijection with the by interpretation at this level and again if you consider for example the as invariant the invariance to be the the invariance of subterminal object you take the bijection at the level of geometric sentences and so on so as again we say that this bijection would be trivial if the two theories are bi-incapable but in this case it's not so there is a toposomal for imaginary so imaginary can be characterized in the toposomal I don't know all the parts well it's just the free topos completion of the coherent syntactic category of the theory so this you can get from the classifying theories okay by taking the coherent object so what we proved when we say that we don't have biinterpretability I mean we don't have coherent biinterpretability in the sense that we prove that the coherent syntactic categories of the two theories are inequivalent and so it made sense to look for other yes if we can just consider a subcategory of we we can we have biinterpretation can we have equivalence yeah so now we can we can obtain also as I say the representation results for this finitely generated algebra in the variety in chunks variety and how we obtain this I say that we obtain this just regarded as at the gradendic topology associated with the theory of preference the algebra regarded as a quotient of the theory of the variety in what we consider in this case and how we obtain this quotient we obtain this quotient just by adding this action to the to the theory of the variety and this action induce a generate the topology that I can write is just you take the you take an algebra you take an element in this algebra and then you consider this this quotient and then you consider finite multi-composition of diagrams of this form and we prove that if you take an algebra that is finitely generated and you compute a diagram like this by choosing at each at each step one of the generators so I can show you the picture here so we choose at each step one of the generator you have that this algebra at the end are perfect and the algebra so and this proved for example that the topology is rigid and so this is another way for proving that the theory of preference the algebra is of a shift type but how we obtain the representation result because at each step the algebra is a product of the two algebra that we obtain as quotient because the elements with respect to we consider the quotient have Boolean element and thanks to a a result that holds in the context of iB algebra it is called push out pull back legma this assure us that this are that a is a product of these two algebras and of course we can apply this lemma at each step and we have that a is a product of the algebras at the end but the question that we want to respond in how is came from this observation we know that the the class of preference the algebra is just the intersection of the class of local in the algebra with the variety with the chunks variety that i that is the variety also generated by this algebra that is chunks algebra that is just an algebras of this form you have an infinite element here and infinite element here where you can imagine an algebra like this and now we prove that this the theory of preference in the algebras of pre-shift type and so the natural question is that what is there is an extrematization for the theory of local in the algebras in an arbitrary sub variety that is a still half pre-shift type and so the next slide slide we want to answer to this question we prove that the globally the theory of local in the algebras is not a pre-shift type but let me just recall the definition local in the algebras are just non-trivial algebras such that for every element you have that is of finite order or the negation of the element is of finite order where the order of the element is just the minimum natural number such that the multiple of the element is equal to one and of course this element cannot be exist and if doesn't exist so you say that the order of the element is infinite and so the question is globally is not a pre-shift type but what doesn't happen if we restrict to the to a proper sub variety and in this case I announced already that the theories of pre-shift type but I want to prove you that this is true so just it just was proved by Komori that every proper sub variety of nv algebra is of this form is generated by a final list of simple nv algebras where simple nv algebras I you can imagine like algebras like this where this is the Moonditch's functor but can be also regarded as subalgebra of the okay okay so just sub algebras of the unit interval where are just of this form you have one there zero one one one m two m and one so this is sm well on the other side you have that this variety can be generated by simple nv algebras and by Komori chain that are algebras of this form okay you can see the the generalization from these algebras and these algebras okay sorry here Komori chain is over this form and so on we prove that even if we change the the generators of the variety they have an invariance so the least common multiple of the ranks of the generators remains the same and we use this number n to axiomatize the the serial local nv algebra in the variety we consider two axiomatization the first axiomatization is the following one that say that every element is is in the radical or it is not in the radical and we prove that the cognitive topology associated to this axiomatization is sub canonical how we prove that because as before you every covering every j1 covering is obtained by is generated by finite multi-composition of diagrams of this form but again this element are Boolean elements and so by again by push up to black lemma we have that this algebra is a product of these two algebras so we are saying that the topology is sub canonical why we need this result we need this result because it's a aloha's to prove that the every every Cartesian sequence that is provable in the theory of local nv algebra in the theory log v is also provable in the theory tv and this since it's very easier to work in the theory log v we use these results to obtain the definability of the radical of the radical for the variety in for the algebra in the variety v but how we obtain this result is just we know that this theory the theory of the variety is in a theory of pre-shift type and so we have the the classifying topos of this is of this form where the classifying topos of the theory of log v is obtained is just a sub-topos of this as the with our objectives had the shifts with respect to j one to the topology j one but the universal model of tv is a model of this form that thanks to the fact that j one is sub canonical is also an object in this topos and can be regarded also as a universal model of this theory and since the interpretation of Cartesian formulas in these two modus in the two difference classifying topos is the same we obtain these we obtain these result this Cartesianization that allows us to prove this definability we prove everything in the theory log v and thanks to the Cartesianization we know that this holds for the theory of tv and in general to the to the to the algebras so just let me go faster we prove that this result about financial presentable models allow assure us that the the condition on the rigidity of the topology associated with the quotient that in general is just a sufficient condition for the theory to be a pre-shift type is also an accessory condition so we just want to prove that the topology associated to the theory log v is a pre-shift type and how we do that we do that by considering another axiomatization that you that we obtained considering the a description of the of the radical class of the algebras and thanks to the fact that these radical classes are compatible with respectable with respect to the mv operation we obtained that the theories of pre-shift have in in what way we consider the topology associated to this second axiomatization that is obtained by finite multi-composition of diagrams of this form and again we for a finite generated algebra we choose at each step the generators and we obtained at the end algebras that are generated by by element that satisfying one of these formulas and thanks to the compatibility condition the algebra generated are again local mv algebras so we we know we prove that these two axiomatization are equal so the gradient topology are the same and this implies the representation results for the finitely generated algebra in the variety v and finally how we obtain the now the Moritz equivalence because we we there is a problem a theorem proved by Dino Esposito in Gerla that represent in this way every local mv algebra in the variety and you see that here we have a group g and a certain k that is the rank of the algebra and I know I didn't talk very much about rank but because I have no time but you can imagine that the theory of l groups is not enough to be Moritz equivalence to the theory of local mv algebras and so we extend the theory of l groups by adding for any for any admissible rank we can say a predicates as a variety predicates that talks about of these ranks and so we obtain this this theory that I call here g ij where the action are this action and the models of this theory in set can be identified with the triples like that so groups constant in a group and k that is almost to regard as the rank of the algebras and now the uh the definition of two functors are very easy I'll just may let me remark the fact that these theories is a pre-shift type because this is the the theorem groups is of pre-shift type here we are doing just Cartesian sequence so this is again a pre-shift type there is a theorem proved by olivia that assure that if you have the action of this form you are still have a theory of pre-shift type so we as is we say we just have to look about the behavior of the models in set and the how we defend the two functions of the categorical equivalence is very easy for any algebra we have the representation by the inola expositor gerla okay I finished the time so okay but you can see by yourself how the functions works and the references for the first two more more each equivalence are these two papers and the first one will appear in forthcoming papers so thank you for your attention I have your question excuse this excuse this very naive question but you said there is an uh uh equivalence between groups and so on and you wrote gamma but how is it different the final gamma no the gamma is defined in a very easy way the game with a group how do you find an algebra yes starting with uh this is the munji munji function it's defined in a very easy way you take uh so gamma from you know elio groups to mv algebras you take a group with a strong unit and then you just consider since these groups are have a have an order as uh and so you can consider just the interval at zero u and this interval you define the operation uh taking as more than the operation uh uh the mv operation on the unit interval zero one so the the sum it's just the term created sum and the negation it's just the u minus x so very very easy definition for the for munji function not so simple is the definition of the of the opposite of the inverse functions and that's how it was presented is not it doesn't seem that it is geometric but we prove in our paper that indeed can be presented in a geometric way also the uh uh the function goings in the other direction the minus is the minus in the group yes yes these are operation in the groups plus and minus yes