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  author = {Vaggione, Diego and S\'anchez Terraf, Pedro},
  title = {Compact factor congruences imply {B}oolean factor congruences},
  journal = {Algebra univers. },
  abstract = {We prove that any variety $\mathcal{V}$ in which every factor congruence is compact has Boolean factor congruences, i.e., for all $A$ in $\mathcal{V}$ the set of factor congruences of A is a distributive sublattice of the congruence lattice of A.},
  volume = {51},
  year = {2004},
  pages = {207--213},
  doi = {10.1007/s00012-004-1857-1},
  zbl = {1087.08001}
  author = {S\'anchez Terraf, Pedro and Vaggione, Diego},
  title = {Varieties with Definable Factor Congruences},
  journal = {Trans. Amer. Math. Soc.},
  volume = {361},
  year = {2009},
  pages = {5061--5088},
  doi = {10.1090/S0002-9947-09-04921-6},
  abstract = {We study direct product representations of  algebras in
varieties. We collect several
conditions expressing that these representations are \emph{definable}
in a first-order-logic sense, among them the concept of  Definable
Factor Congruences (DFC). The main results are that DFC  is a Mal'cev
property and that it  is equivalent to all other conditions
formulated; in particular we prove that $\mathcal{V}$ has DFC if  and only if 
$\mathcal{V}$ has $\vec{0}$ \& $\vec{1}$ and \emph{Boolean Factor Congruences}.  We also  obtain an explicit
 first order definition $\Phi$ of the kernel of the canonical projections via the terms 
associated to the Mal'cev condition for DFC, in such a manner it is preserved
by taking direct products and direct factors. The main tool is the use
of \emph{central elements,} which are a generalization of
both central idempotent elements in rings with identity and neutral
complemented elements in a bounded lattice.},
  zbl = {1223.08001},
  mrnumber = {2515803}
  author = {S\'anchez Terraf, Pedro},
  title = {Existentially definable factor congruences},
  journal = {Acta Scientiarum Mathematicarum (Szeged)},
  volume = {76},
  number = {1--2},
  year = {2010},
  pages = {49--53},
  url = {https://bit.ly/3cBZP9f},
  eprint = {0906.4722},
  abstract = {A variety $\mathcal{V}$  has \emph{definable factor congruences} if and only if
factor congruences can be defined by a first-order formula $\Phi$ having
\emph{central elements} as parameters. We prove that if $\Phi$ 
can be chosen to be existential, factor congruences in every
algebra of $\mathcal{V}$ are compact.},
  zbl = {1274.08028}
  author = {S\'anchez Terraf, Pedro},
  title = {Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids},
  journal = {Order},
  volume = {25},
  number = {4},
  year = {2008},
  pages = {377--386},
  doi = {10.1007/s11083-008-9101-9},
  abstract = {We study varieties with a term-definable poset structure, \emph{po-groupoids}. It is known
that connected posets have the \emph{strict refinement property}
(SRP). In a previous work by Vaggione and the author it is proved that 
semidegenerate varieties with the SRP have definable factor congruences
and if the similarity type is finite, directly indecomposables are
axiomatizable by a set of first-order  sentences. We obtain such a set
for semidegenerate varieties of connected po-groupoids 
and show its quantifier complexity is bounded in general.},
  zbl = {1168.08005}
  title = {Unprovability of the logical characterization of bisimulation},
  journal = {Information and Computation},
  volume = 209,
  number = 7,
  issue = 7,
  pages = {1048--1056},
  year = 2011,
  issn = {0890-5401},
  doi = {10.1016/j.ic.2011.02.003},
  url = {http://www.sciencedirect.com/science/article/pii/S0890540111000691},
  author = {S{\'a}nchez Terraf, Pedro},
  keywords = {Labelled Markov process},
  keywords = {Probabilistic bisimulation},
  keywords = {Modal logic},
  keywords = {Nonmeasurable set},
  abstract = {We quickly review \emph{labelled Markov processes
  (LMP)} and provide a counterexample showing that in general
measurable spaces, event
bisimilarity and state bisimilarity differ in LMP. This shows that 
the logic in the work by Desharnais does not 
characterize state bisimulation in non-analytic measurable
spaces. Furthermore we show that, under current foundations of
Mathematics, such logical characterization is unprovable for spaces
that are projections of a coanalytic set. Underlying this construction
there is a proof that stationary Markov processes over general
measurable spaces do not have semi-pullbacks.},
  zbl = {1216.68196}
  author = {S{\'a}nchez Terraf, Pedro},
  title = {{{B}oolean factor congruences and property $(*)$}},
  language = {English},
  journal = {Int. J. Algebra Comput. },
  volume = {21},
  number = {6},
  pages = {931-950},
  year = {2011},
  doi = {10.1142/S021819671100656X},
  abstract = {{Summary: A variety ${\cal V}$ has Boolean factor congruences (BFC) if
    the set of factor congruences of any algebra in ${\cal V}$ is a distributive
    sublattice of its congruence lattice; this property holds in rings with unit
    and in every variety which has a semilattice operation. BFC has a prominent
    role in the study of uniqueness of direct product representations of
    algebras, since it is a strengthening of the refinement property.
    provide an explicit Mal'tsev condition for BFC. With the aid of this
    condition, it is shown that BFC is equivalent to a variant of the
    definability property $(*)$, an open problem in {\it R. Willard}'s work [J.
    Algebra 132, No.~1, 130--153 (1990; Zbl 0737.08006)].}},
  keywords = {{Boolean factor congruences; strict refinement property; definability;
    preservation by direct factors}},
  classmath = {{*08B05 (Equational logic in varieties of algebras)
  zbl = {1228.08003}
  author = {D'Argenio, Pedro R. and S\'{a}nchez Terraf, Pedro  and Wolovick, Nicol\'{a}s},
  title = {Bisimulations for non-deterministic labelled {M}arkov processes},
  journal = {Mathematical Structures in Comp. Sci.},
  issue_date = {February 2012},
  volume = {22},
  number = {1},
  issue = 1,
  month = feb,
  year = {2012},
  issn = {0960-1295},
  pages = {43--68},
  numpages = {26},
  url = {http://dx.doi.org/10.1017/S0960129511000454},
  doi = {10.1017/S0960129511000454},
  acmid = {2139685},
  publisher = {Cambridge University Press},
  address = {New York, NY, USA},
  abstract = {  We extend the theory of labeled Markov processes with
  \emph{internal} nondeterminism, a fundamental concept for the further
  development of a process theory with abstraction on nondeterministic
  continuous probabilistic systems.
  We define \emph{nondeterministic labeled Markov processes (NLMP)}
  and provide three definition of bisimulations: a bisimulation
  following a traditional characterization, a \emph{state} based
  bisimulation tailored to our ``measurable'' non-determinism, and an
  \emph{event} based bisimulation.
  We show the relation between them, including that the largest state
  bisimulation is also an event bisimulation.
  We also introduce a variation of the Hennessy-Milner logic that
  characterizes event bisimulation and that is sound w.r.t.\ the other
  bisimulations for arbitrary NLMP.
  This logic, however, is infinitary as it contains a denumerable
  We then introduce a finitary sublogic that characterize all
  bisimulations for image finite NLMP whose underlying measure space
  is also analytic. Hence, in this setting, all notions of
  bisimulation we deal with turn out to be equal.
  Finally, we show that all notions of bisimulations are different in
  the general case.  The counterexamples that separate them turn to be
  \emph{non-probabilistic} NLMP.},
  zbl = {1234.68316}
  author = {S{\'a}nchez Terraf, Pedro},
  title = {Bisimilarity is not {B}orel},
  abstract = {We prove that the relation of bisimilarity between countable labelled transition systems is $\Sigma_1^1$-complete (hence not Borel), by reducing the set of non-wellorders over the natural numbers continuously to it. 
This has an impact on the theory of probabilistic and nondeterministic processes over uncountable spaces, since logical characterizations of bisimilarity (as, for instance, those based on the unique structure theorem for analytic spaces) require a countable logic whose formulas have measurable semantics. Our reduction shows that such a logic does not exist in the case of image-infinite processes.},
  ee = {http://arxiv.org/abs/1211.0967},
  keywords = {Mathematics - Logic, Computer Science - Logic in Computer Science, 03B70, 03E15, 28A05, F.4.1, F.1.2},
  journal = {Mathematical Structures in Computer Science},
  issn = {1469-8072},
  doi = {10.1017/S0960129515000535},
  zbl = {1377.68150},
  url = {http://journals.cambridge.org/article_S0960129515000535},
  pages = {1265--1284},
  number = {7},
  month = {10},
  year = {2017},
  volume = {27}
  author = {S{\'a}nchez Terraf, Pedro},
  title = {Factor Congruences in Semilattices},
  journal = {Revista de la {U}ni\'on {M}atem\'atica {A}rgentina},
  volume = {52},
  number = {1},
  year = {2011},
  ee = {http://arxiv.org/abs/0809.3822v2},
  eprint = {0809.3822},
  keywords = {semilattice, direct factor, factor congruence, generalized direct sum, generalized ideal},
  pages = {1--10},
  url = {http://inmabb.criba.edu.ar/revuma/pdf/v52n1/v52n1a03.pdf},
  abstract = {We characterize factor congruences in semilattices by using generalized notions of order ideal and of direct sum of ideals. When the semilattice has a minimum 
(maximum) element, these generalized ideals turn into ordinary (dual) ideals.},
  zbl = {1242.06006}
  author = {Doberkat, Ernst-Erich and S{\'a}nchez Terraf, Pedro},
  title = {Stochastic Nondeterminism and Effectivity Functions},
  journal = {Journal of Logic and Computation},
  archiveprefix = {arXiv},
  eprint = {1405.7141},
  primaryclass = {cs.LO},
  keywords = {stochastic effectivity function, non-deterministic labelled Markov process, state bisimilarity, coalgebra},
  year = 2015,
  doi = {10.1093/logcom/exv049},
  abstract = {This paper investigates stochastic nondeterminism on continuous state spaces by relating nondeterministic kernels and stochastic effectivity functions to each other. Nondeterministic kernels are functions assigning each state a set o subprobability measures, and effectivity functions assign to each state an upper-closed set of subsets of measures. Both concepts are generalizations of Markov kernels used for defining two different models: Nondeterministic labelled Markov processes and stochastic game models, respectively. We show that an effectivity function that maps into principal filters is given by an image-countable nondeterministic kernel, and that image-finite kernels give rise to effectivity functions. We define state bisimilarity for the latter, considering its connection to morphisms. We provide a logical characterization of bisimilarity in the finitary case. A generalization of congruences (event bisimulations) to effectivity functions and its relation to the categorical presentation of bisimulation are also studied.}
  author = {Areces, Carlos and Campercholi, Miguel and Penazzi, Daniel and S{\'a}nchez Terraf, Pedro},
  title = {{The Lattice of Congruences of a Finite Linear Frame}},
  journal = {Journal of Logic and Computation},
  archiveprefix = {arXiv},
  eprint = {1504.01789},
  primaryclass = {math.LO},
  keywords = {Mathematics - Logic, Computer Science - Logic in Computer Science, 03B45 (Primary), 06B10, 06E25, 03B70 (Secondary), F.4.1, F.1.2},
  doi = {10.1093/logcom/exx026},
  zbl = {06981765},
  pages = {2653--2688},
  volume = 27,
  issue = 8,
  abstract = {Let $\mathbf{F}=\left\langle F,R\right\rangle $ be a finite Kripke frame. A
congruence of $\mathbf{F}$ is a bisimulation of $\mathbf{F}$ that is also an
equivalence relation on F. The set of all congruences of $\mathbf{F}$ is a
lattice under the inclusion ordering. In this article we investigate this
lattice in the case that $\mathbf{F}$ is a finite linear frame. We give
concrete descriptions of the join and meet of two congruences with a nontrivial
upper bound. Through these descriptions we show that for every nontrivial
congruence $\rho$, the interval $[\mathrm{Id_{F},\rho]}$ embeds into the
lattice of divisors of a suitable positive integer. We also prove that any two
congruences with a nontrivial upper bound permute.},
  year = 2017,
  month = apr,
  adsurl = {http://adsabs.harvard.edu/abs/2015arXiv150401789A},
  adsnote = {Provided by the SAO/NASA Astrophysics Data System}
  author = {Pachl, Jan and S{\'a}nchez Terraf, Pedro},
  title = {Semipullbacks of labelled {M}arkov processes},
  journal = {Logical Methods in Computer Science},
  volume = 17,
  number = 2,
  year = 2021,
  month = apr,
  archive = {arXiv},
  eprint = {1706.02801},
  primaryclass = {math.PR},
  keywords = {Mathematics - Probability, Computer Science - Logic in Computer Science, 28A35, 28A60, 68Q85, F.4.1, F.1.2},
  adsurl = {http://adsabs.harvard.edu/abs/2017arXiv170602801P},
  adsnote = {Provided by the SAO/NASA Astrophysics Data System},
  doi = {10.23638/LMCS-17(2:3)2021},
  url = {https://lmcs.episciences.org/7361},
  abstract = {A  \emph{labelled Markov process (LMP)} consists of a measurable
  space $S$ together with an indexed family of Markov kernels from $S$
  to itself. This structure has been used to model probabilistic
  computations in  Computer Science, and one of the main problems in
  the area is to define and decide whether two LMP $S$ and $S'$ ``behave
  the   same''. There are two natural categorical definitions of
  sameness of behavior: $S$  and $S'$  are \emph{bisimilar}
  if there exist  an LMP  $ T$ and measure preserving maps
  forming a diagram of the shape 
  $ S\leftarrow  T  \rightarrow{S'}$; and  they
  are \emph{behaviorally equivalent} 
  if there exist  some $ U$ and  maps forming a dual  diagram
   $ S\rightarrow  U  \leftarrow{S'}$.

  These two notions differ for general measurable spaces but Edalat
  proved that they coincide for analytic Borel spaces, showing that
  from every 
  diagram  $ S\rightarrow  U  \leftarrow{S'}$ one can obtain a
  bisimilarity diagram as above. Moreover, the resulting square of
  measure preserving maps is commutative (a \emph{semipullback}).

  In this paper, we extend Edalat's result to measurable spaces $S$
  isomorphic to a  universally measurable subset of 
  a Polish space with the trace of the Borel  $\sigma$-algebra, using a
  version of Strassen's theorem on 
  common extensions of finitely additive measures.}
  author = {Caicedo, Xavier and  Campercholi, Miguel and  Kearnes, Keith A. and  S{\'a}nchez Terraf, Pedro and  Szendrei, {\'A}gnes and Vaggione, Diego},
  year = 2021,
  title = {Every minimal dual discriminator variety is minimal as a quasivariety},
  journal = {Algebra universalis},
  month = {Apr},
  day = 29,
  volume = 82,
  number = 2,
  pages = 36,
  abstract = {Let $\dagger$ denote the following property of a variety $\mathcal{V}$: \emph{Every subquasivariety of $\mathcal{V}$ is a variety}. In this paper, we prove that every idempotent dual discriminator variety has property $\dagger$ . Property $\dagger$ need not hold for nonidempotent dual discriminator varieties, but $\dagger$ does hold for \emph{minimal} nonidempotent dual discriminator varieties. Combining the results for the idempotent and nonidempotent cases, we obtain that every minimal dual discriminator variety is minimal as a quasivariety},
  issn = {1420-8911},
  doi = {10.1007/s00012-021-00715-8},
  url = {https://doi.org/10.1007/s00012-021-00715-8}
  title = {The {Z}hou Ordinal of Labelled {M}arkov Processes over Separable Spaces},
  author = {Moroni, Martín Santiago  and S\'anchez Terraf, Pedro},
  journal = {arXiv e-prints},
  month = may,
  year = 2020,
  eprint = {2005.03630},
  archive = {arXiv},
  primaryclass = {cs.LO},
  note = {Accepted for publication at the Review of Symbolic Logic},
  abstract = {There exist two notions of equivalence of behavior between states of a
Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The
first one can be considered as an appropriate generalization to continuous
spaces of Larsen and Skou's probabilistic bisimilarity, while the second one is
characterized by a natural logic. C. Zhou expressed state bisimilarity as the
greatest fixed point of an operator $\mathcal{O}$, and thus introduced an
ordinal measure of the discrepancy between it and event bisimilarity. We call
this ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\mathbb{S})$.
When $\mathfrak{Z}(\mathbb{S})=0$, $\mathbb{S}$ satisfies the Hennessy-Milner
property. The second author proved the existence of an LMP $\mathbb{S}$ with
$\mathfrak{Z}(\mathbb{S}) \geq 1$ and Zhou showed that there are LMPs having an
infinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb{S}$
over separable metrizable spaces having arbitrary large countable
$\mathfrak{Z}(\mathbb{S})$ and that it is consistent with the axioms of
$\mathit{ZFC}$ that there is such a process with an uncountable Zhou ordinal.}

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