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In this paper we study some fragments without implications of the (Hilbert)full Lambek logic $\mathbf{HFL}$ and also some fragments without implicationsof some of the substructural extensions of that logic. To do this, we performan algebraic analysis of the Gentzen systems defined by the substructuralcalculi $\FL_\sigma$. Such systems are extensions of the full Lambek calculus$\FL$ with the rules codified by a subsequence, $\sigma$, of the sequence $ew_l w_r c$; where $e$ stands for \emph{exchange}, $w_l$ for \emph{leftweakening}, $w_r$ for \emph{right weakening}, and $c$ for \emph{contraction}.We prove that these Gentzen systems (in languages without implications) arealgebraizable by obtaining their equivalent algebraic semantics. All theseclasses of algebras are varieties of pointed semilatticed monoids and they canbe embedded in their ideal completions. As a consequence of these results, wereveal that the fragments of the Gentzen systems associated with the calculi$\FL_\sigma$ are the restrictions of them to the sublanguages considered, andwe also reveal that in these languages, the fragments of the external systemsassociated with $\FL[\sigma]$ are the external systems associated with therestricted Gentzen systems (i.e., those obtained by restriction of$\FL_\sigma]$ to the implication-less languages considered). We show that allthese external systems without implication have algebraic semantics but theyare not algebraizable (and are not even protoalgebraic). Results concerningfragments without implication of intuitionistic logic without contraction werealready reported in Bou et al.(2006): On two fragments with negation andwithout implication of the logic of residuated lattices. Archive forMathematical Logic 45(5) and in Adill\'on et al. (2007): On threeimplication-less fragments of t-norm based fuzzy logics. Fuzzy Sets and Systems158(23).

On Fragments without Implications of both the Full Lambek Logic and some of its Substructural Extensions