In this paper we study some fragments without implications of the (Hilbert) full Lambek logic $\mathbf{HFL}$ and also some fragments without implications of some of the substructural extensions of that logic. To do this, we perform an algebraic analysis of the Gentzen systems defined by the substructural calculi $\FL_\sigma$. Such systems are extensions of the full Lambek calculus $\FL$ with the rules codified by a subsequence, $\sigma$, of the sequence $e w_l w_r c$; where $e$ stands for \emph{exchange}, $w_l$ for \emph{left weakening}, $w_r$ for \emph{right weakening}, and $c$ for \emph{contraction}. We prove that these Gentzen systems (in languages without implications) are algebraizable by obtaining their equivalent algebraic semantics. All these classes of algebras are varieties of pointed semilatticed monoids and they can be embedded in their ideal completions. As a consequence of these results, we reveal that the fragments of the Gentzen systems associated with the calculi $\FL_\sigma$ are the restrictions of them to the sublanguages considered, and we also reveal that in these languages, the fragments of the external systems associated with $\FL[\sigma]$ are the external systems associated with the restricted Gentzen systems (i.e., those obtained by restriction of $\FL_\sigma]$ to the implication-less languages considered). We show that all these external systems without implication have algebraic semantics but they are not algebraizable (and are not even protoalgebraic). Results concerning fragments without implication of intuitionistic logic without contraction were already reported in Bou et al.(2006): On two fragments with negation and without implication of the logic of residuated lattices. Archive for Mathematical Logic 45(5) and in Adill\'on et al. (2007): On three implication-less fragments of t-norm based fuzzy logics. Fuzzy Sets and Systems 158(23).

New types of systems of fuzzy relation inequalities and equations, called weakly linear, have been recently introduced in [J. Ignjatovi\'c, M. \'Ciri\'c, S. Bogdanovi\'c, On the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations, Fuzzy Sets and Systems 161 (2010) 3081--3113.]. The mentioned paper dealt with homogeneous weakly linear systems, composed of fuzzy relations on a single set, and a method for computing their greatest solutions has been provided. This method is based on the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations. Here we adapt this method for computing the greatest solutions of heterogeneous weakly linear systems, where the unknown fuzzy relation relates two possibly different sets. We also introduce and study quotient fuzzy relational systems and establish relationships between solutions to heterogeneous and homogeneous weakly linear systems. Besides, we point out to applications of the obtained results in the state reduction of fuzzy automata and computing the greatest simulations and bisimulations between fuzzy automata, as well as in the positional analysis of fuzzy social networks.

Recently, two types of simulations (forward and backward simulations) and four types of bisimulations (forward, backward, forward-backward, and backward-forward bisimulations) between fuzzy automata have been introduced. If there is at least one simulation/bisimulation of some of these types between the given fuzzy automata, it has been proved that there is the greatest simulation/bisimulation of this kind. In the present paper, for any of the above-mentioned types of simulations/bisimulations we provide an effective algorithm for deciding whether there is a simulation/bisimulation of this type between the given fuzzy automata, and for computing the greatest one, whenever it exists. The algorithms are based on the method developed in [J. Ignjatovi\'c, M. \'Ciri\'c, S. Bogdanovi\'c, On the greatest solutions to certain systems of fuzzy relation inequalities and equations, Fuzzy Sets and Systems 161 (2010) 3081-3113], which comes down to the computing of the greatest post-fixed point, contained in a given fuzzy relation, of an isotone function on the lattice of fuzzy relations.

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